1. Introduction
Axisymmetric linear magnetic mirror traps with neutral beam injection (NBI) can potentially become the basis for a volumetric fusion neutron source (FNS) (Anikeev et al. Reference Anikeev, Bagryansky, Beklemishev, Ivanov, Kolesnikov, Korzhavina, Korobeinikova, Lizunov, Maximov and Murakhtin2015a; Bagryansky et al. Reference Bagryansky, Chen, Kotelnikov, Yakovlev, Prikhodko, Zeng, Bai, Yu, Ivanov and Wu2020), applicable to the fields of thermonuclear material science and to fusion–fission hybrid reactors (Yurov & Prikhod'ko Reference Yurov and Prikhod'ko2014; Yurov & Prikhodko Reference Yurov and Prikhodko2016).
The gas-dynamic trap (GDT) in the Budker Institute of Nuclear Physics (Ivanov & Prikhodko Reference Ivanov and Prikhodko2017) experimentally supports the idea that an axisymmetric linear magnetic mirror trap can be developed into a FNS. In GDT plasmas neutrons are born in D–D reactions between deuterium nuclei originating from NB injection, which is the main plasma heating system of the GDT.
‘Fast’ or ‘sloshing’ ions are formed via charge exchange between injected neutral beams and cold dense target plasma ions, confined in the gas-dynamic regime (Mirnov & Riutov Reference Mirnov and Riutov1979). The fast ion energy confinement time is much shorter than their angular scattering time, and their mean free path far exceeds the length of the trap. As a result, their velocity distribution function is far from Maxwellian and possesses a loss cone, i.e. they are confined to the so-called ‘kinetic’ regime. The sloshing ions have their axial pressure and density profile maxima in the vicinity of two turning points where the magnetic field strength is twice that of in the centre of the trap. These are also the region where the neutron flux peaks, which is useful for designing an FNS (Simonen et al. Reference Simonen, Anikeev, Bagryansky, Beklemishev, Ivanov, Lizunov, Maximov, Prikhodko and Tsidulko2010; Ivanov & Prikhodko Reference Ivanov and Prikhodko2017). However, in kinetic confinement regimes with non-Maxwellian velocity distribution functions, kinetic instabilities tend to develop, enhancing the scattering of sloshing ions into the loss cone. This scattering also smooths out the fast ions axial pressure profile, decreasing the neutron flux at its maxima points, thus decreasing the efficiency of an FNS.
The drift cyclotron loss cone (DCLC) instability (Post Reference Post1987) is one of the most likely instabilities to occur under the magnetic mirror trap conditions. The DCLC instability can be considered as a flute-like plasma potential perturbation with a small transverse wavelength, i.e. $k_{\perp } \rho _{i}\geqslant 1$
(where $k_{\perp }$
and $\rho _{i}$
are the transverse wave vector component and ion gyroradius, respectively), which propagates azimuthally in the ion diamagnetic drift direction. This instability can evolve from drift waves that appear in the presence of a radial plasma density profile gradient inherent to magnetic mirror confinement systems. Drift waves interact with ions and grow in amplitude since the presence of the loss cone and transverse non-uniformity of the plasma create ion velocity distribution functions with regions of positive derivative.
A stability criterion for the DCLC instability can be formulated using the parameter $\rho _{i}/r_{{\rm pl}}$
, where $r_{{\rm pl}}$
is the plasma column radius (Post & Rosenbluth Reference Post and Rosenbluth1966; Kotelnikov, Chernoshtanov & Prikhodko Reference Kotelnikov, Chernoshtanov and Prikhodko2017). The threshold value of this parameter is $\rho _{i}/r_{{\rm pl}} \sim 10^{-2}$
if the effects of the limited trap length and the finite relative pressure $\beta$
are taken into consideration and the low-energy end of the distribution is unpopulated (Tang, Pearlstein & Berk Reference Tang, Pearlstein and Berk1972). The value of this parameter in the GDT for fast ions is too high ($0.1 \leqslant \rho _{i}/r_{{\rm pl}} \leqslant 1$
), and thus this condition is not fulfilled. Therefore, the GDT plasma is unstable against the DCLC mode in the fully kinetic regime. However, the kinetic regime is considered to be more reasonable for implementation in FNS projects based on the GDT as opposed to the gas-dynamic regime, since the kinetic regime provides longer energy confinement time (Yurov & Prikhodko Reference Yurov and Prikhodko2016) and a higher D–T reaction neutron flux (Bagryansky et al. Reference Bagryansky, Chen, Kotelnikov, Yakovlev, Prikhodko, Zeng, Bai, Yu, Ivanov and Wu2020).
There are some particular properties of the DCLC instability that allow us to distinguish it from other kinetic instability types. First, the base frequency of the instability is close to the ion cyclotron frequency; the instability can also develop at harmonics of its base frequency. Second, although the assumption $k_{\perp } \rho _{i}\geqslant 1$
was used to simplify the dispersion equation, it was shown in several papers (Post & Rosenbluth Reference Post and Rosenbluth1966; Kotelnikov et al. Reference Kotelnikov, Chernoshtanov and Prikhodko2017) that this condition is valid for the DCLC instability branches with the highest increment values. Thus, an inequality $|m|=|k_{\varphi }r_{{\rm pl}}| \geqslant 1$
for the azimuthal wavenumber $m$
follows from that condition as well as from the comparability of the ion gyroradius and transverse plasma length. Third, the wave propagates in the direction of the ion diamagnetic drift. These properties were experimentally observed in several mirror devices: 2XII (Simonen Reference Simonen1976), 2XIIB (Turner Reference Turner1977; Turner, Powers & Simonen Reference Turner, Powers and Simonen1977), TMX-U (Berzins & Casper Reference Berzins and Casper1987), MIX 1 (Koepke et al. Reference Koepke, McCarrick, Majeski and Ellis1986) and LAMEX (Ferron & Wong Reference Ferron and Wong1984). The radial structure of plasma potential perturbations has to be determined by the radial density distribution of warm ions, i.e. ions distributed isotropically in velocity space, which was confirmed in experiments conducted on MIX 1 (Koepke et al. Reference Koepke, McCarrick, Majeski and Ellis1986) and LAMEX (Ferron & Wong Reference Ferron and Wong1984).
The DCLC instability is inherently sensitive to the features of the ion velocity distribution function. Theory predicts that the DCLC instability can be suppressed by partially filling the loss cone with warm ions. The efficiency of this method has been proven in experiments on PR-6 (Kanaev Reference Kanaev1979), 2XIIB (Coensgen et al. Reference Coensgen, Cummins, Logan, Molvik, Nexsen, Simonen, Stallard and Turner1975), TMX-U (Berzins & Casper Reference Berzins and Casper1987) and MIX 1 (Koepke et al. Reference Koepke, McCarrick, Majeski and Ellis1986), in which warm ion fractions in plasma, at which the DCLC was stabilised, were evaluated at 5 %–20 %.
As noted in Kotelnikov & Chernoshtanov (Reference Kotelnikov and Chernoshtanov2018), there exists a so-called ‘spectral rule’: the warm ions effectively stabilise the DCLC instability if their cyclotron harmonics coincide with those of the fast ions. The experiments on PR-6 facility (Kanaev Reference Kanaev1979) confirmed this by demonstrating that less than 5 % of warm hydrogen was enough to prevent the DCLC instability from forming in hydrogen plasma, while in the case of warm argon its density had to be comparable to the density of the entire plasma.
Implementing a FNS requires use of two-species plasmas (most importantly, D–T plasmas) that are DCLC stable. However, the stabilisation of FNS plasmas by addition of warm ions requires larger amounts of particles compared with single-species plasmas, shifting the plasma away from the kinetic confinement to the gas-dynamic regime and decreasing the output of a neutron source. This issue can be mitigated by using a two-species warm plasma as the target for the neutral beams’ capture, so that the first species serves as the necessary fuel component while the second provides the DCLC stabilisation.
In order to test this stabilisation technique we conducted experiments where the hot ion population comprised deuterium and the warm ion population hydrogen. In these circumstances, the cyclotron harmonics of warm ions would only overlap with the even deuterium harmonics, so the DCLC instability was expected to form at odd deuterium harmonics, primarily at the first one. The instability was expected to be completely suppressed when a certain amount of warm deuterium ions was added.
The objectives for this study were to:
(i) identify instability types by their properties;
(ii) suppress the instability in the GDT by using a two-species target plasma (in our case, D–H);
(iii) determine the ratio of isotope densities in the mixture at which the plasma is DCLC stable.
The paper is organised as follows. Section 2 provides a brief description of the GDT, employed diagnostics and signal processing methods as well as a detailed description of the experimental routine and the method used for preparing target plasma and controlling its isotopic composition. Section 3 contains the results obtained on several qualitatively different ion-cyclotron instabilities, one of which we attributed to the DCLC type, the experimentally determined ratio of deuterium to hydrogen in a mixed target which suppresses the DCLC instability and measurements of additional longitudinal energy losses due to the instability with estimates of energy lost per ion. The findings of the study are concluded in § 4.
2. Description of the experiment
2.1. The GDT device
The GDT is an axisymmetric magnetic mirror device. The GDT (figure 1a) consists of a central solenoid with two magnetic mirror coils at its ends, with the maximum mirror ratio of $R_{\max } = B_{\max }/B_{\min } = 34$
. The plasma heating system consists of 8 NBIs with 24 keV deuterium beams inclined at $45^\circ$
to the GDT axis, supplying up to 5 MW of power in total. A comprehensive description of the GDT facility is provided in Ivanov & Prikhodko (Reference Ivanov and Prikhodko2017).
Figure 1. (a) Scheme of the GDT and its axial magnetic field profile: 1a, 1b – pulsed gas valves; 2 – arc discharge plasma generator; 3 – NB injectors; 4a, 4b – limiters; 5a, 5b, 5c – diamagnetic loops; 6 – azimuthal array of the magnetic probes; 7 – longitudinal array of the magnetic probes; 8 – azimuthal array of the electrostatic probes; 9 – dispersion interferometer's line of sight (LOS); 10 – spectrometer's LOS plane; 11 – diagnostic plasma absorber; 12 – LOS plane of the Thomson scattering diagnostic; (b) layout of the particle and energy flux diagnostics on the western absorber plate (not to scale); black circles mark the pyroelectric bolometers, red circles – the ion current probes; (c) photo of the azimuthal array of seven magnetic probes and a longitudinal array of the magnetic probes; (d) azimuthal array of electrostatic probes; (e) arrangement of the probes for radial correlation measurement.
The experimental routine proceeded as follows. The gas valves were opened in advance and the vacuum vessel was filled with the prepared gas mixture; the valves were closed at the ‘0’ ms moment of the experimental routine. The arc plasma generator in the eastern expander tank fired from 0.4 to 4.2 ms and produced a plasma with a temperature of a few eV, which then passed through the magnetic mirror and together with the gas comprised a cold dense plasma. This plasma served as a target for neutral deuterium beams injected from 3.8 to 8.6 ms; the charge exchange between target plasma ions and neutral beams resulted in formation of the fast deuteron population. The trapped NBI power was kept approximately constant throughout the experimental campaign (${\approx }1.8$
MW). At 4.4 ms electric potential was applied to the limiters to stabilise magnetohydrodynamic (MHD) instabilities during discharge via the ‘vortex confinement’ method (Beklemishev et al. Reference Beklemishev, Bagryansky, Chaschin and Soldatkina2010; Bagryansky et al. Reference Bagryansky, Anikeev, Beklemishev, Donin, Ivanov, Korzhavina, Kovalenko, Kruglyakov, Lizunov and Maximov2011).
Two different plasma components formed in the GDT during the discharge. The first was the warm dense plasma with the Maxwellian velocity distribution. This component, consisting of both ions and electrons, was confined in the gas-dynamic regime (Mirnov & Riutov Reference Mirnov and Riutov1979) due to its high collision frequency. The second component was the fast anisotropic ion population with an essentially non-equilibrium velocity space distribution, similar to the velocity distribution of the injected beams. This property arises from the fact that the fast ion energy confinement time, determined by the ion–electron collisions, is much shorter than their angular scattering time. The fast ion motion is governed by conservation of the magnetic moment and energy; as the result, fast ions move in the region between the points where the local mirror ratio $R = B/B_{\min } = 2$
, i.e. the turning points, where their axial density profile peaks.
The diamagnetic pressure (mostly exerted by fast ions) was measured using three diamagnetic loops (figure 1a). One of the loops was located at the GDT centre and the other two were at the positions with the values of the $R=1.32$
and $R=2$
. Magnetic field perturbations were registered by azimuthal and longitudinal magnetic probe arrays. Electric potential perturbations were measured with an azimuthal array of electrostatic probes. The dispersion interferometer (Lvovskiy & Solomakhin Reference Lvovskiy and Solomakhin2011) was used to measure the linear electron density along the LOS passing through the trap axis in its central plane. The Thomson scattering diagnostic (Lizunov et al. Reference Lizunov, Berbasova, Khilchenko, Kvashnin, Puryga, Sandomirsky and Zubarev2023) was used to record electron density and temperature at several points along the radius in the central plane. Additionally, the particle and energy flux measurement system, located on the western plasma absorber, was used to measure longitudinal energy and particles fluxes (figure 1b). The system consisted of 16 sets of probes, each set containing a pyroelectric bolometer and an ion current probe (individual probe designs are described in Soldatkina et al. Reference Soldatkina, Maximov, Prikhodko, Savkin, Skovorodin, Yakovlev and Bagryansky2020).
2.2. Magnetic probes
The spatial structure of the instabilities was studied using an azimuthal array of high-frequency magnetic probes (figure 1c). Each probe consisted of a single square-shaped turn of copper wire with an area of $1\ {\rm cm}^2$
, insulated by Teflon, oriented azimuthally and spaced apart at steps of $7.5^\circ$
. One of the probes had an additional radially oriented turn, so that the polarisation of the magnetic perturbation in the ($r, \varphi$
) plane could be determined. The probes were capable of recording signals with frequencies of up to 100 MHz. The signals were recorded with an ADC at the rate of $500\ {\rm MSamples}\ {\rm s}^{-1}$
.
The array was installed at the location with the local mirror ratio $R=1.44$
perpendicularly to the GDT axis. Each coil's centre was located 21 cm away from the axis, as placing them any closer leads to perceptible fast ions losses. The radius of the magnetic surface touching the limiters and serving as the warm plasma boundary was 13.7 cm. Since the plasma in the GDT centre can have a $\beta$
of up to ${\approx }10\,\%$
(with an obtained maximum of approximately 60 % at a turning point demonstrated in Bagryansky et al. Reference Bagryansky, Anikeev, Beklemishev, Donin, Ivanov, Korzhavina, Kovalenko, Kruglyakov, Lizunov and Maximov2011), the probes were capable of measuring the properties of the plasma instabilities through the associated magnetic field perturbations despite the remoteness of the probes.
Additionally, a longitudinal array consisting of similar magnetic probes was used in this study. Six probes measuring the axial component of magnetic field perturbations were distributed over the length of the region between the centre and the western fast ion turning point (figure 1a,c).
2.3. Electrostatic probes
An array of electrostatic probes was used to corroborate the magnetic probe data. The array (figure 1d) was installed right in front of the western limiter, at the local mirror ratio of $R=4.33$
. Placing it far away from the fast ion confinement region allowed us to move these probes close to the plasma edge without disrupting the plasma. Each of the eight probes consisted of a molybdenum wire of 0.2 mm diameter shielded by a copper tube grounded to the vacuum chamber. A thin silica glass tube insulated the wire from the shield, and another glass tube insulated the copper shield from contact with the plasma and the metal base of the array. A 5 mm long Mo-wire tip protruded beyond the shield and served as the antenna. The probe tips were located at $10^\circ$
steps along the arc with an 8 cm radius, which is the radius of the plasma column at this location.
For another experiment where we measured the phase shift between the probe signals at different radii (figure 1e) four of those eight probes were arranged into two pairs. Because of the high heat flux that the probes would experience in the GDT, instabilities could not be investigated across the entire plasma cross-section and measurements had to be conducted at the plasma edge where the probes could withstand the heat. In each pair one probe touched the plasma edge (probes numbered 3 and 5, figure 1e) and the other was slightly removed from the edge radially (probes 8 and 7 at distances of 1 and 2 cm from their counterparts, respectively). They were also spaced from each other along the GDT axis by 12 mm, which was accounted for during evaluation of the perturbation‘s radial wavelength (with limitations specified in § 3.4).
There was only one ADC capable of recording electric and magnetic probe signals, so either magnetic or electrostatic probe signals were being recorded at a time.
2.4. Data processing technique
An electromagnetic perturbation $\delta U$
can be represented as a superposition of azimuthal modes m due to the cylindrical geometry of a magnetic mirror trap
The analysis of a perturbation's azimuthal structure is based on the assumption that a single azimuthal mode of the instability dominates during its development.
Instabilities in discharges were observed as sequences of self-correlated independent oscillation bursts with their correlation times ranging from 5 to $100\ \mathrm {\mu } {\rm s}$
(figure 5a,b). An instability's spatial structure can be constant during time periods under $100\ \mathrm {\mu } {\rm s}$
, and a new instability with the same frequency but different spatial properties can form while the previous one decays. Hence, the decision was made to process probe data in short time windows of either 5 or $10\ \mathrm {\mu } {\rm s}$
.
Every window underwent the following processing before being subjected to FFT. First, the offset and the linear slope were subtracted from the signal data. After that, the window was multiplied by the Hanning window and extended fourfold by appending zeroes to obtain better spectral resolution. Then, having obtained Fourier images $U_{x}(f)$
, $U_{y}(f)$
of the simultaneous windows $u_{x}(t)$
, $u_{y}(t)$
of probes $x$
and $y$
, the complex cross-spectrum was calculated as
where the cross-power $P_{x,y}(f)$
denotes the frequency domain of the signals’ congruence and the cross-phase $\varTheta _{x,y}(f)$
is the phase difference between the two signals. This procedure was applied to signals from all pairs of adjacent probes ($n$
in total), giving $n-1$
cross-powers and cross-phases.
The cross-power magnitude can be used to identify time–frequency regions with reliable signal correlation according to the criterion from Scargle (Reference Scargle1982, expression 18). This criterion is applied in § 3.2 for analysis of electrostatic probe signals. These time–frequency regions were combined for $n-1$
pairs of adjacent probes and the cross-phases were averaged across the resulting region. For electrostatic probes such values of the azimuthal wavenumber $m$
were chosen for which the standard error did not exceed $\Delta m = 0.5$
; whether the procedure worked correctly was determined by visual inspection of the obtained data.
Magnetic probes collected much more data, therefore these data had to be processed automatically. The time–frequency regions had to be chosen differently: in each time interval a more narrow vicinity of the spectral maximum was chosen than the criterion allows. As the result, the data samples taken from the magnetic probe data were even more statistically reliable. The phase shift $\varTheta _{x,y}$
between remote magnetic probes (i.e. probes that have other probes between them) $x$
and $y$
in the azimuthal array is given by $\varTheta _{x,y}=\sum _{i=x}^{y-1}\varTheta _{i,i+1},$
where $\varTheta _{i,i+1}$
is the cross-phase of adjacent (to each other) probes i and ${i+1}$
. If only one azimuthal mode is present, the dependence of these phase shifts on the angular distance between the probes $x$
and $y$
is linear, and $m$
is given by the slope of the line. This way the wavenumber $m$
can be determined for each 5 or $10\ \mathrm {\mu } {\rm s}$
observation window of the instability evolving.
During the magnetic probe data processing, we used two criteria to determine whether a mode was present in a discharge. The first criterion was that the coefficient of determination of the fit line $R^2$
had to exceed 0.85. The second criterion was that all values of the phase shift between adjacent probes had to have the same sign (a single value, however, was allowed to differ). The described criteria were not applicable to the case of $m=0$
, since the phase shifts were small and had random signs.
Bicoherence squared $b^2$
was estimated to determine nonlinear coupling between the fluctuation components with frequencies $f_{1}$
and $f_{2}$
using
where the angular brackets $\langle {\dots} \rangle$
denote averaging over signal windows.
2.5. Target plasma composition control
The isotopic composition of the target plasma was adjusted by mixing $D_2$
and $H_2$
and using the mixture in both the initial gas target and in the arc discharge plasma gun. The mixture was prepared in corresponding reservoirs and its composition was controlled via pressure gauges.
An auxiliary series of experiments was conducted to determine if the composition of the target plasma corresponded exactly to the composition of the gas mixture used. A spectrometer and an optical system similar to the one described in Lizunov et al. (Reference Lizunov, Den Hartog, Donin, Ivanov and Prikhodko2011) as a part of the MSE diagnostic were used to measure the percentage of deuterium in the plasma during the discharge. The optical system (10 in figure 1a) collected radiation along four chords in the plane perpendicular to the GDT axis and at different distances from the axis. The time of exposure start was varied while the exposition time of 0.5 ms remained constant. The registered radiation belonged to a spectrum range including ${H_\alpha }$
and ${D_\alpha }$
lines, which were well resolved by the spectrometer. The ratio between the ${D_\alpha }$
line intensity (spectral density integral in the vicinity of the spectral line, $I({D_\alpha })$
) and the total intensity of both spectral lines $I$
was taken to be the percentage of deuterium in the plasma. This follows from the assumption that the excitation rate coefficients for radiation emission at ${H_\alpha }$
and ${D_\alpha }$
frequencies are the same. This assumption is justified due to the same energy structure of deuterium and hydrogen (neglecting the isotopic shift) and the high speed of electrons far exceeding that of the ions; however, due to that high speed both warm and fast ions contributions are accounted for, both of which are considerable.
Figure 2(a) shows the ${D_\alpha }$
contribution to the total intensity depending on the chord-to-axis distance projected on the central plane. The results are plotted for several different exposure start times. The deuterium fraction in plasma was constant across the plasma column radius (with a relative error not exceeding 7 %) up to the limiter edge projection onto the central plane (black dashed line in figure 2a), but dropped during a discharge. It can be seen more clearly in figure 2(b) that the deuterium content decreased by approximately 18 % during the NBI. Such decline could be caused by hydrogen accumulating in the plasma because of its desorption from the first wall and the limiters as well as because of unnoticed leakage of the water vapour into the vacuum chamber. Bearing in mind that the used gas mixture contained $36\pm 3\,\%$
of deuterium, as determined from partial pressures, we concluded that at the start of the NB injection the deuterium fraction in the plasma was the same as in the prepared gas mixture, and that the deuterium content was uniform across the viewed cross-section.
Figure 2. The ${D_\alpha }$
contribution to the total spectral density integral in the vicinity of spectral lines for both ${H_\alpha }$
and ${D_\alpha }$
in the $36\pm 3\,\%$
$D_{2}$
target plasma: (a) temporal dependence for different chord radius central plane projections; the dashed line shows the limiter edge projection onto the central plane; (b) temporal dependence of $I({D_\alpha })/I$
during NB injection for two particular radii (triangles) and the mean values over all the measured radii (circles).
3. Results and discussion
3.1. Ion-cyclotron instabilities
An instability with a frequency of approximately 2.2 MHz (figure 3a) was observed in pure hydrogen target plasmas as well as in plasmas with specific hydrogen–deuterium ratios (more on the ratios in § 3.3). The frequency of the instability was close to the ion-cyclotron (IC) frequency of deuterium in the GDT centre (2.74 MHz). Varying the magnetic field strength proportionally changed the instability frequency, confirming its IC nature. The discrepancy $\Delta f$
between the instability and IC frequencies could arise from the local magnetic field strength decreasing due to plasma diamagnetism, as suggested by Turner et al. (Reference Turner, Powers and Simonen1977). However, in our case this is unlikely since during a discharge the instability frequency changes only slightly while the diamagnetic flux changes significantly (see figures 3b, 4a, 4c) and $\Delta f$
exceeds the expected value of $\Delta f/f=1-\sqrt {1-\beta }<0.05$
for the maximum value of $\beta \lesssim 0.1$
in the central plane. Moreover, the Doppler shift of the instability frequency due to the $\boldsymbol {E}\times \boldsymbol {B}$
plasma rotation (${\sim }10$
kHz) can be neglected. The discrepancy could also be explained by the fact that non-local analysis of DCLC modes in a cylindrical geometry (Ferraro et al. Reference Ferraro, Littlejohn, Sanuki and Fried1987) predicts the DCLC to appear at a frequency significantly lower than that of the IC, but proper comparison would require taking into account longitudinal inhomogeneity, the radial distribution of fast ions (which differs from pure Gaussian) and finite beta and finite Larmor radius effects.
Figure 3. Data from a discharge with a 2.2 MHz instability (no. 51035): (a) spectrogram of a magnetic probe signal during an instability evolving in a discharge with a mixed hydrogen–deuterium target ($\langle n_el\rangle =(2.9\pm 0.4)\times 10^{14}\ {\rm cm}^{-2}$
, 46 % deuterium mixture). The black line marks the IC frequency of deuterium in the GDT centre; (b) diamagnetic flux signals from the diamagnetic loops; (c) power and ion current fluxes from a bolometer and ion current probe pair on the plasma absorber.
Figure 4. Instabilities evolving in the discharges with a pure deuterium target under (a) moderate plasma density $\langle n_el\rangle = (2.9\pm 0.3)\times 10^{14}\ {\rm cm}^{-2}$
and (b) high plasma density $\langle n_el\rangle = (4.9\pm 0.3)\times 10^{14}\ {\rm cm}^{-2}$
; (c,d) corresponding diamagnetic flux signals.
Figure 4(a) illustrates the instability at a frequency of approximately 4.5 MHz that formed in pure deuterium plasmas. As in the previous case, its frequency was proportional to the magnetic field strength, i.e. it also had an IC nature. This instability's frequency was close to both the second harmonic of the deuterium IC frequency in the GDT centre and the IC frequency associated with the magnetic field at the turning points ($R=2$
), which was equal to 5.48 MHz.
The 4.5 MHz instability seemed to originate as both an independent instability at the turning point and as a result of nonlinear coupling of the 2.2 MHz instability near the trap centre. Figure 5(c) shows the spectra of three probe (7 on figure 1a) signals that were obtained in a plasma discharge with a pure hydrogen target. If the two instabilities are flute-like in nature, the probes at different locations must be probing the same wave; thus, by the amplitude of the signal their point of origin can be traced. The highest spectral peak at 2.2 MHz belonged to the probe closest to the centre ($R=1.19$
), while the highest spectral peak at 4.5 MHz belonged to the probe near the turning point ($R=2.1$
). The signal from the probe located near the azimuthal array ($R=1.47$
) had the weakest spectrum peaks at both aforementioned frequencies. This can be interpreted as the 4.5 MHz instability principally arising in the vicinity of turning points independently from the 2.2 MHz instability. The instability evolving near the turning point has been observed in Berzins & Casper (Reference Berzins and Casper1987), where its localisation was reliably determined by skewing the neutral beams. It has been demonstrated in Ferron & Wong (Reference Ferron and Wong1984) that several DCLC modes with various frequencies can originate at different points where particle–wave resonance is effective, in particular, in regions with $\boldsymbol {\nabla } B \approx 0$
. On the other hand, $b^2$
(calculated from an azimuthal magnetic probe data at $R=1.44$
, figure 5d) peaks at $f_{1}=f_{2}\approx 2.2$
MHz, suggesting that it is the nonlinear coupling of the 2.2 MHz instability that is the reason of the additional 4.5 MHz spectrum component formation.
Figure 5. (a) Envelope of the raw magnetic probe signal ($R=1.44$
, black line) and envelopes of this signal passed through several band-passes that correspond to the base frequency and its second and third harmonics. (b) Time dependence of the phase in the range of 2–2.5 MHz which includes the base frequency; (c) signal spectra of three magnetic probes installed at different local mirror ratio points (7 in figure 1a); (d) bicoherence squared calculated from a probe signal ($R=1.44$
) by averaging 200 time windows with $23\ \mathrm {\mu } {\rm s}$
duration each.
Whenever instabilities formed, simultaneous changes in diamagnetic flux signals were observed (figures 3b and 4c). The drops in the $R=2$
loop signal coincided with rises in other loops signals, especially the central loop ($R=1$
). Additionally, simultaneous spikes in power and ion fluxes on the plasma absorber were recorded (figure 3c). This suggests that the instability caused scattering of fast ions (since they are the ones that primarily contribute to the diamagnetic flux signals) and redistributed the diamagnetic pressure along the trap axis, smoothing the maxima of the axial pressure profile.
The instabilities were observed in different discharges with the linear electron density $\langle n_{e}l\rangle$
, measured with the dispersion interferometer (9 in figure 1a), ranging from $2.3\times 10^{14}$
to $3.6\times 10^{14}\ {\rm cm}^{-2}$
. For a typical discharge with a hydrogen target characterised by a time-averaged (from 5 to 9 ms) value of $\langle n_{e}l\rangle = (2.9\pm 0.2)\times 10^{14}\ {\rm cm}^{-2}$
(from here on out, the provided $\langle n_{e}l\rangle$
values denoted the mean $\pm$
standard deviation on the time interval), figures 6(b) and 6(c) show radial profiles of electron temperature $T_e$
and density $n_e$
obtained with the Thomson scattering diagnostic before and during the instability. During the discharge $T_{e}$
values increased due to NB heating, while the $n_{e}$
profile, that initially peaked near the axis, flattened. Although we cannot infer how the instability affected the plasma parameters specifically and how the plasma density profile change influenced the instability, these data allowed us to determine the conditions in which it exists in the GDT.
Figure 6. (a) Spectrogram of a 2.2 MHz instability registered with an electrostatic probe. Vertical grey lines denote time points at which radial profiles of (b) electron temperature and (c) density were measured.
Increasing the linear plasma density to ${\sim }5\times 10^{14}\ {\rm cm}^{-2}$
by operating the second gas valve allowed us to completely avoid the 4.5 MHz instability in pure deuterium plasmas and almost entirely suppress the 2.2 MHz instability in pure hydrogen plasmas. This observation is in line with what we expect from the DCLC instability, i.e. its being suppressed by increasing the warm ion density, thus filling the loss cone.
High-frequency instabilities were also observed in mixed deuterium–hydrogen plasmas of both moderate and high densities. A pure deuterium discharge is shown in figure 4(b), in which we observed a few harmonics ranging from approximately 25 to 80 MHz in steps of approximately 5 MHz, which was the deuterium IC frequency at the fast ion turning points. These harmonics were also observed in microwave radiation power spectra during the collective Thomson scattering (CTS) diagnostic implementation experiments on the GDT (Shalashov et al. Reference Shalashov, Gospodchikov, Khusainov, Lubyako, Solomakhin and Yakovlev2022). Nonetheless, this instability did not influence the diamagnetic flux (figure 4d) noticeably and thus we did not investigate it further during this study.
An increase in the target plasma density, in combination with the proper tuning of the gas puffing duration, allowed us to raise the power captured by the plasma from the NBI (up to ${\approx }2.5$
MW) as well as the diamagnetic flux (over ${\approx }70$
kMx in the central plane) in a pure deuterium target discharge. Thus, with increased neutral gas puffing, $\beta$
exceeded the stability threshold and the Alfvén ion-cyclotron instability (AICI) formed, which caused fast ions to scatter and limited the diamagnetic flux growth rate. Besides, increased density of the target plasma contributed to the AIC instability's arising because it decreases the Alfvén wavelength and thereby weakens the effect of the longitudinal magnetic field inhomogeneity on stabilisation (Watson Reference Watson1980). At linear density values of $6.0-6.7\times 10^{14}\ {\rm cm}^{-2}$
the 2.2 and 4.5 MHz instabilities were suppressed and only AIC instability developed (more in § 3.4). These values were obtained in pure deuterium targets; as this study was mostly focused on DCLC instability, the influence of target plasma composition on AICI was not investigated.
As shown previously in Zaytsev et al. (Reference Zaytsev, Anikeev, Bagryansky, Donin, Kovalenko, Korzhavina, Lizunov, Lozhkina, Maximov and Pinzhenin2013, Reference Zaytsev, Anikeev, Bagryansky, Donin, Korobeinikova, Korzhavina, Kovalenko, Lizunov, Maximov and Pinzhenin2014), the AIC instability in the GDT develops at the base frequency of 1.3 MHz or less and with the azimuthal wavenumber of $m=1$
. In our experiments, using electrostatic probes we observed the same AIC instability being self-correlated during its entire lifetime (approximately 1 ms) and disappearing after NBI ended. Determination of its azimuthal wavenumber, which turned out to be $m=1$
, allowed us to verify the signal processing procedure described previously in § 2.4. The following sections show that spatial properties of the 2.2 MHz instability were different from those of the AICI, therefore, the features of the observed AIC instability clearly show that the instabilities at 2.2 and 4.5 MHz have a different nature. Further on, we will refer to the 2.2 MHz and 4.5 MHz oscillations as the DCLC instability since their properties satisfy its description.
3.2. Azimuthal structure of the DCLC instability
For discharges with a pure hydrogen target and the strong 2.2 MHz instability, electrostatic probe signals were processed in steps of $10\ \mathrm {\mu } {\rm s}$
with $30\ \mathrm {\mu } {\rm s}$
windows. Within each window instabilities were distinguished in cross-power spectrum using the detection threshold from Scargle (Reference Scargle1982, expression 18). Phase shifts in the common time–frequency domain among all pairs of the adjacent probes were averaged to plot colour maps displaying normalised azimuthal wavenumbers $m=k_{\varphi }r_{pl}$
within this domain (figure 7a,b).
Figure 7. (a,b) Time dependencies of azimuthal wavenumbers (distinguished by colour) of the 2.2 MHz instability for two discharges with different plasma densities; (c) envelope curves of electrostatic probe signals band-pass filtered between 1.9 and 2.6 MHz (smoothed over $30\ \mathrm {\mu } {\rm s}$
) and (d) time evolution of the linear electron density in these discharges; (e,f) distributions of azimuthal wavenumbers for two groups of discharges: (e) the low density $\langle n_{e}l\rangle = (2.2\pm 0.2)\times 10^{14} {\rm cm}^{-2}$
case, (f) the higher density $\langle n_{e}l\rangle = (3.6\pm 0.2)\times 10^{14}\ {\rm cm}^{-2}$
case.
It can be seen that the way the instability develops depended on the target plasma density: at low density $\langle n_{e}l\rangle = (2.2\pm 0.2)\times 10^{14}\ {\rm cm}^{-2}$
the instability started as high-amplitude (figure 7c) narrow-band oscillations with apparent positive azimuthal wavenumbers in the 1–5 range and continued as wider band low-amplitude oscillations with widely spread wavenumbers. At higher density $\langle n_{e}l\rangle = (3.6\pm 0.2)\times 10^{14} {\rm cm}^{-2}$
the instability immediately started as low-amplitude, wide-band oscillations with absolute values of its wavenumbers close to unity. To support these observations, distributions over two groups of discharges, related to either lower density or higher density, are shown in figures 7(e) and 7(f), respectively. Note that the change in how the instability develops in the low density case is not connected to variations in $\langle n_{e}l\rangle$
during a discharge (figure 7d). This change could be caused by modification of the radial warm ion density profile during a discharge that was expected for the DCLC instability and has been observed in Koepke et al. (Reference Koepke, McCarrick, Majeski and Ellis1986) and Ferron & Wong (Reference Ferron and Wong1984).
When the instability began to develop and had a high amplitude at low plasma densities, its wavenumbers were positive, which means that it propagated in the ion diamagnetic drift direction. This feature is compatible with properties of the DCLC instability. However, when the instability evolved at a lower amplitude (especially at high plasma densities), its wavenumbers became partly negative and smaller in absolute values. These features mean a larger scale of the instability spatial structure and propagation in the electron drift direction. The latter is not typical for the DCLC instability, although it has been observed earlier by Turner (Reference Turner1977).
Swift changes in azimuthal wavenumbers (especially between 5.6 and 6.6 ms in figure 7a) agree with our understanding that that the instability evolves as a sequence of separate short oscillations (also seen in figure 5a,b). This might be explained by a build-up of a drift wave by a group of fast ions leading to scattering of the said ions and thus the resonance conditions’ breaking. As the result, the instability decayed, but a new one formed in its place due to the NBI supplying new fast ions. However, the instability did not necessarily scatter the resonant ions. Another mechanism of an instability's intermittence (bursting) was proposed in Ferron & Wong (Reference Ferron and Wong1984). The authors observed the DCLC instability, whose frequency, wavelength, phase velocity and the radial profile of the amplitude all satisfied the resonance conditions and the requirement for the azimuthal wavenumbers to be integer. Since plasma parameters changed, it could be that a dominant saturated mode with a given wavenumber no longer satisfied the requirements, and another one, that did satisfy them, grew to became dominant. The amplitude bursting took place simultaneously at different azimuthally and radially located points, so the plasma was unstable in its entire cross-section. On the contrary, on the same LAMEX machine (Leikind et al. Reference Leikind, Zwi, Dimonte and Wong1985) it was shown that, besides the aforementioned reasons, the instability bursting can be caused by its spatial localisation, with bursts arising when a spatially localised mode passes by the probes as a part of a large-scale convective plasma motion.
In our case, all of these effects could be responsible for the instability bursting. However, it is the scattering of the resonant ions and hence the resonance breaking that is most likely to be the main reason of the bursting, due to high estimated energy of the lost ions (§ 3.5). It is difficult to determine if the other effects were responsible for bursting because of the small size of both probe arrays ($45^\circ$
and $70^\circ$
) and because of the ongoing NB injection, whereas in the reported experiments (Ferron & Wong Reference Ferron and Wong1984; Leikind et al. Reference Leikind, Zwi, Dimonte and Wong1985) the instability developed during free decay of plasma.
3.3. The DCLC in plasmas of different compositions
Ion-cyclotron instabilities at frequencies between 2.2 and 5 MHz were observed in discharges with pure hydrogen and deuterium target plasmas as well as in experiments with varying isotopic target plasma compositions. The experimental series with magnetic probe measurements was characterised by $\langle n_el \rangle$
between $2.3\times 10^{14}$
and $3.6\times 10^{14}\ {\rm cm}^{-2}$
(with a mean value of $\langle n_el \rangle = (3\pm 0.4)\times 10^{14}\ {\rm cm}^{-2}$
). The edge values of this range correspond to the two target plasma densities used in wavenumber measurements with electrostatic probes. Thus, it would be reasonable to expect similar values of the azimuthal wavenumbers measured by magnetic probes, and indeed, they were mainly distributed between 2 and 6 for both measurements taken by magnetic probes and by electrostatic probes in low density plasmas (figures 8b and 7a). However, negative wavenumbers were almost invisible to the magnetic probes, which could be due to the stricter selection criterion used in their data processing compared with electrostatic probes (see § 2.4).
Figure 8. Properties of the instability at: (a–c) the base frequency of approximately 2.2 MHz, (d–f) the frequencies that are non-multiples of the base frequency: 4.2 MHz (red), 3.8 MHz (black) and 3 MHz (blue) versus deuterium content in the plasma target. For both cases: (a, d) the frequency distributions; (b, e) the normalised azimuthal wavenumber $m$
distributions; (c, f) the distributions of the magnetic perturbation polarisation's $\theta _{r-\varphi }$
normalised by ${\rm \pi}$
. All histograms are normalised by total numbers of measurements (in (a) and (d) plots).
The 2.2 MHz instability formed in every discharge until the deuterium content in the target plasma exceeded 54 %. This instability had a narrow enough spectrum to process the probe signals using the technique described in § 2.4. These signals were processed in windows of $5\ \mathrm {\mu } {\rm s}$
to obtain temporal dependencies of the azimuthal wavenumber as well as the perturbations polarisation $\theta _{r-\varphi }$
in the plane almost normal to the magnetic field. The latter is the phase difference between signals from $r$
- and $\varphi$
-oriented coils of the magnetic probe. The results are shown in figure 8(a–c). The frequency distributions in figure 8a have a wide spread. The reason for this is that these distributions were made using the data from the entire duration of the discharges, thus they were affected by the instability frequency's drifting during the discharge.
The normalised azimuthal wavenumber $m$
ranged from 1 to 7 for all target plasma compositions with values in the 2–5 range dominating in most cases (figure 8b) with no specific time dependence. Thus, we cannot conclude that the azimuthal structure of the instability depended on deuterium content. The unstable waves propagated in the ion diamagnetic drift direction regardless of the target composition.
The polarisation angle distribution was almost uniform and its values were mostly positive in pure hydrogen target plasmas (figure 8c). However, as deuterium content increased, a well-defined maximum became sharper and gravitated to ${\rm \pi} /2$
. Since amplitudes of the $r$
- and $\varphi$
-oriented coils signals near 2.2 MHz are comparable, this means that the registered perturbations had a near-circular polarisation in the radial plane. This, however, is likely not a feature of the instability itself, but could be an effect caused by the probe remoteness from the core plasma, as magnetic perturbations in cold magnetised plasmas must be polarised circularly. The plasma column radius (taken to be the radius of the magnetic surface that touches the inner edge of the radial limiters) was approximately 14 cm in the plane where the probes were located, whereas the probes were located 21 cm away from the axis and immersed in the cold peripheral plasma.
In pure hydrogen target plasmas only the 2.2 MHz instability and its harmonics were observed; however, even with a small added amount of deuterium, instabilities at other frequencies, non-multiples of the base frequency, began to develop. Figure 8(d–f) demonstrates the properties of such instabilities grouped by their frequencies. Most instability azimuthal wavenumber distributions peaked at $m = 3$
or $m = 4$
. Such instabilities arose mainly before the 2.2 MHz instability and lasted less time (see figure 6(a) at 6 ms for instance). Azimuthal wavenumbers did not vary significantly while the instabilities developed. The instabilities propagated in the ion diamagnetic drift direction and the magnetic perturbations polarisation distributions had their maxima even closer to ${\rm \pi} /2$
than in the 2.2 MHz case.
The deuterium content threshold under which the 2.2 MHz instability developed was between 54 % and 65 %. Of all DCLC-like instabilities, only the 4.5 MHz frequency and its harmonics remained when the deuterium content exceeded 65 %. Unfortunately, we were unable to determine the azimuthal wavenumbers of the 4.5 MHz instability since the phase shifts between the magnetic probes differed from each other significantly in their values and even in their signs. This could be due to different spatial modes’ superposing as well as due to superposition of modes from both turning points.
3.4. Radial structure of the AIC and DCLC instabilities
Using pairs of radially displaced electrostatic probes (§ 2.3, figure 1e), we measured the radial phase shifts of DCLC and AIC instabilities on the plasma periphery.
The AIC instability amplitude decreased significantly with the distance from the plasma column (figure 9a), however, phase differences between the signals of the third and eighth probes, as well as between the fifth and seventh probes, were discernible. Different axial positions of radially distributed probes could contribute to the phase shift despite the AICI longitudinal wavelength $\lambda _{\parallel }=136\pm 4$
cm (Chen et al. Reference Chen, Bagryansky, Zeng, Zou, Zhang, Wang, Jia, Zhang, Dong and Zha2022) far exceeding the distance between the paired probes along the axis (12 mm). The phase difference due to axial distance was $\varTheta _{\parallel }=0.06$
rad; it was subtracted from the measured differences between the probe signals. The radial wavenumber $k_r$
could be obtained as the average of $(\varTheta _{3, 8}-\varTheta _{\parallel })/1\ {\rm cm}^{-1}$
and $(\varTheta _{5, 7}-\varTheta _{\parallel })/2\ {\rm cm}^{-1}$
, however, we chose a different method instead to highlight the discrepancy between phase differences for two probe pairs. We plotted the distributions of these phase differences normalised by the distances between the probes in pairs (figure 9b). The distributions show that we cannot consider the radial structure of the instability to be a single harmonic, i.e. ${\sim }\exp (\mathrm {i}k_{r}r)$
, so $k_{r}$
should be understood as an inverted scale of perturbations. Both distributions were fitted with Gaussian curves to obtain the mean and the variance, and these values were used to calculate $k_r$
as the weighted mean: $k_r = 0.11\pm 0.03\ {\rm cm}^{-1}$
. The corresponding AICI radial scale is $\lambda _{r} = 57\pm 16$
cm, which significantly exceeds the plasma column radius. Presuming that the same value of $\lambda _r$
is true for inside the core plasma as well, we can conclude that the AIC instability was uniform within the entire plasma cross-section, as proposed in Tsang & Smith (Reference Tsang and Smith1987).
Figure 9. (a) The AIC instability registered with electrostatic probes (numbered according to figure 1e); (b) phase differences normalised by radial spacings between the probes.
Figure 10(a) shows the amplitude drop of the 2.2 MHz instability with distance from the plasma edge. Assuming this to be the DCLC instability, we can consider its $k_\parallel \ll k_\perp$
, neglecting the phase shifts between the probes in pairs 5–7 and 3–8 (figure 1e) due to their different axial positions. Figures 10(b) and 10(c) present distributions of phase differences between radially spaced probes in the same discharges as in figures 7(e), 7(f). Figure 10(b) shows results from several discharges with low linear electron density of $(2.2\pm 0.2)\times 10^{14}\ {\rm cm}^{-2}$
. Despite double the difference in distances between the probes, both distributions have almost the same mean values of 1.47 and 1.56 rad. Using these values and their corresponding variances, $k_{r}=0.90\pm 0.34\ {\rm cm}^{-1}$
and the corresponding radial scale of the DCLC instability at low densities $\lambda _{r} = 7.0\pm 2.6$
cm were derived. This value is approximately equal to the plasma column radius at the axial position of the probes. Similar distributions obtained for discharges with higher linear density $(3.6\pm 0.2)\times 10^{14}\ {\rm cm}^{-2}$
(figure 10c) had closer-to-zero mean values, so the corresponding $k_{r}$
had to be smaller. The same was valid for $m$
(figure 7f); thus we can conclude that the transverse scale of the instability tended to grow as the plasma density increased.
Figure 10. (a) The 2.2 MHz instability registered with electrostatic probes; phase differences for (b) the low plasma density $\langle n_{e}l\rangle = (2.2\pm 0.2)\times 10^{14}\ {\rm cm}^{-2}$
and (c) the higher plasma density $\langle n_{e}l\rangle = (3.6\pm 0.2)\times 10^{14}\ {\rm cm}^{-2}$
cases.
3.5. Scattered ion energy estimation
Using the plasma absorber diagnostics, we attempted to estimate the energy of lost ions. First, we examined the distribution of power and ion current over the area of the absorber. Plotting the fluxes averaged over each probe wing ($45^{\circ }, 135^{\circ }, 225^{\circ }$
and $315^{\circ }$
) showed that the fluxes were independent of a probe's azimuthal angle (figure 11a), as expected due to the axial symmetry of the GDT. There was, however, a noticeable difference in power fluxes at different plasma absorber radii (figure 11b). When the fluxes were averaged over each ring of probes (i.e. over the same radius of the absorber), the highest peaks were observed at the radius of 28.2 cm, which, projected along the magnetic field lines in the GDT, corresponds to the radius of 7.3 cm in the central plane. The second highest peaks were observed at the absorber radius of 16.6 cm (4.3 cm in the central plane respectively); as for outer radii, no such sharp rises were observed. The ion current signals at these radii showed a similar behaviour, with the peaks at $r_{2}=28.2$
cm being the most prominent.
Figure 11. Examples of (a) azimuthal and (b) radial power flux distributions in a plasma discharge.
A numerical simulation of collisionless ion dynamics was carried out to obtain the spatial resolution of the bolometers. In this simulation, the ions were placed inside the bolometer and allowed to fly apart isotropically, backtracking to their possible origins at the centre of the trap. Velocities and flight times of ions that reached the central plane were investigated as well as the radii at which they entered the central plane. The simulation showed that the spatial resolution did not exceed 3.3 cm in the radial direction for ions with 10 keV energies and the direct flight time $\tau _{{\rm flight}}$
from the bolometers to the central plane was found to be approximately $6\ \mathrm {\mu } {\rm s}$
.
Presuming the flux increases to be caused by instabilities, we attempted to verify if those losses could be the fast ions scattered from the centre of the trap. Assuming that the peaks of the bolometer signals were caused by scattered fast ion losses alone, we attempted to estimate their energy from the available data. To isolate these losses from the normal operational losses (henceforth referred to as ‘background’ losses), for both the power and ion current flux signals we fit straight lines into the short time period $\tau _{{\rm approx}}$
before the instability formed (figure 12a). These straight lines, meant to simulate the background losses in the absence of the instabilities, were then extrapolated into the regions with the instabilities and deducted from the corresponding signals, bringing the power and ion flux signals to the same baseline. After that, we synchronised the flux signals to a magnetic probe signal with a 1.6–2.4 MHz band-pass filter applied programmatically. The band-passing was performed with the aim of tracking the appearance and growth of the 2.2 MHz instability. Having noticed that the magnetic probe, power and current signals had a similar shape shifted in time, we estimated the time delay $\tau _{{\rm delay}}$
between the instability start and the ions reaching the absorber as the distance between the maxima of the signals (figure 12b). However, $\tau _{{\rm delay}}$
is partially produced by the bolometer's response time of $15\ \mathrm {\mu } {\rm s}$
. Additionally, we estimated the instability pulse length time $\tau _{{\rm inst}}$
, which was used in the next step of the analysis.
Figure 12. Visual guideline for the procedure used to determine the average energy of lost fast ions: (a) time period $\tau _{{\rm approx}}$
before the instability was fit with a straight line to separate the instability-driven losses from background losses; (b) data were re-plotted with the fit lines subtracted, the magnetic probe signal was overlaid, $\tau _{{\rm inst}}$
and $\tau _{{\rm delay}}$
were determined; (c) graph of $P_{{\rm ring} 2}/J_{{\rm ring} 2}$
was plotted and used to estimate the average energy of the ions lost due to the instability; additional blue lines correspond to the energy of injected neutral beams $E_{{\rm inj}}$
and the average energy of the background losses $E_{{\rm back}}$
calculated over $\tau _{{\rm approx}}$
.
By dividing the resultant backgroundless power flux by the backgroundless ion current (figure 12b), we obtained the time dependency of the energy per ion estimate $E_{{\rm ion}}$
(figure 12c). If the bolometer response time was taken into account, even higher estimates of $E_{{\rm ion}}$
would be obtained. Note that the values of the resultant dependency lie below the injected neutral beam's energy $E_{{\rm inj}} = 24$
keV and above the background value of $E_{{\rm back}} = 1.6\pm 0.2$
keV, obtained by averaging the quotient of power by ion current (before subtracting the fit lines) over $\tau _{{\rm approx}}$
. The resultant signal was then averaged over $\tau _{{\rm inst}}$
, shifted forward from the instability start time by $\tau _{{\rm delay}}$
. As the result, we obtained an estimate for the average energy per instability-scattered ion in the discharge. The procedure was repeated for 19 plasma discharges to obtain a range of average energy values. Some discharges, however, had several subsequent instability pulses in them. In these cases, we only analysed the first one, as we were unsure how the first pulse would affect the background flow for the following pulses.
In all the processed discharges, the estimated energy of instability-scattered ions ranged from 4 to 16 keV, with a mean and the standard deviation of $10\pm 4$
keV. These values significantly exceeded the energies of the background losses, comprised mostly of warm ions $E_{{\rm back}}$
. In addition, we determined that the average $\tau _{{\rm delay}}$
equalled $41\pm 3\ \mathrm {\mu } {\rm s}$
.
We used the obtained results to estimate the transverse energy of the ions appropriate to fulfil the DCLC characteristic condition $k_{\perp }\rho _{i}\geqslant 1$
. The majority of lost ions reached the plasma absorber from the central plane radius of $r_{2}=7.3\pm 1.5$
cm with energies in the range of $E_{{\rm ion}}=10\pm 4$
keV. One can expect that the DCLC instability has the highest amplitude at this radius. Using the measured value of $m=4\pm 2$
we obtained $k_{\varphi }=m/r_{2}=0.56\pm 0.30\ {\rm cm}^{-1}$
. Thereby, the condition $k_{\varphi }\rho _{i}\geqslant 1$
for the most unstable DCLC branches was satisfied for deuterons with transverse energies exceeding the threshold value ranging from 0.4 to 4.6 keV (depending on the value of $k_{\varphi }$
). Taking $k_{r}$
into account (projected to the central plane) gave a value for $k_{\perp }=0.72\pm 0.26\ {\rm cm}^{-1}$
and somewhat different estimations of transverse ion energy: 0.3–1.5 keV.
We also attempted to determine the region of the velocity space from which the ions were lost. Assuming that the instability (characterised by $f=2.2$
MHz and $m=4\pm 2$
) resonated with these ions, we evaluated their transverse velocities using
and then obtained the pitch angle $\theta _{{\rm ion}}$
using $\sin ^2\theta _{{\rm ion}}=m_{D} v^2_{\perp } / 2E_{{\rm ion}}$
. The pitch angles of ions leaving the confinement region determined using this method turned out to be in the range of $\theta _{{\rm ion}}=14.7\pm 8.6^\circ$
, which is in line with the loss cone angle of $\theta _{lc} = \arcsin (1/\sqrt {R_{\max }}) = 9.9^\circ$
. The value of $\theta _{{\rm ion}}$
could be even closer to the loss cone angle if contribution of $k_{r}$
to $k_{\perp }$
was used in the evaluation of $v_{\perp }$
. Moreover, the average $\tau _{{\rm delay}}$
between the moments when the DCLC bursts with the highest amplitude and when the power losses spikes for the first time equalled ${\approx }40\ \mathrm {\mu } {\rm s}$
. According to simulations, a 10 keV deuteron reaches the absorber in $\tau _{{\rm flight}}\sim 6\ \mathrm {\mu } {\rm s}$
. In addition, our pyroelectric bolometer response time is $5\ \mathrm {\mu } {\rm s}$
to register 50 % of an instant power flux pulse and $15\ \mathrm {\mu } {\rm s}$
to register 90 %. Given the period of a single bounce oscillation ${\sim }10\ \mathrm {\mu } {\rm s}$
, the ions had to oscillate at least 2 or 3 times before reaching the plasma absorber. It follows that the lost ions had to already be near the loss cone in the phase space to get scattered by the instability in the collisionless regime. Therefore, it is possible that the DCLC-like IC instability developed by interacting with the ions located near the loss cone in the velocity space and eventually scattered them after a short time.
However, this method did not allow us to estimate energies of scattered ions when applied to the AIC instability. In regimes with the AIC instability the target plasma density and its longitudinal particle losses were several times higher than the corresponding density and losses in regimes with the DCLC instability, so the current of lost fast ions could not be discerned well from the background current. Additionally, there was an uncertainty in determining $\tau _{{\rm delay}}$
. Direct measurements of the lost ions energy spectrum when AICI evolves were conducted in Anikeev et al. (Reference Anikeev, Bagryansky, Zaitsev, Korobeinikova, Murakhtin, Skovorodin and Yurov2015b).
3.6. Total losses caused by instabilities
Energy and particle fluxes measured by probes on the diagnostic plasma absorber were integrated over its area under the assumption of uniformity of the fluxes within the sectors surrounding each probe. As the result, the total power $P_{{\rm abs}}$
and the total ion current $J_{{\rm abs}}$
of plasma leaving the trap through the western mirror were obtained. As seen in figure 13, both the 2.2 MHz DCLC instability (a) and the AIC instability (b) provoked an almost twofold temporary increase of total power losses and a relatively weak but discernible increase of the total ion current at the moment they arise. After the initial spike, the longitudinal energy losses, assumed to be caused by the DCLC instability, became almost constant but higher by ${\approx }25\,\%$
than they were before the instability onset. In contrast, longitudinal power losses, caused by the AIC instability, increased for approximately 0.4 ms and later returned to the levels that would seemingly be observed in the absence of the instability. This can be interpreted as fast ions with pitch angle close to the loss cone angle being lost. These losses are driven by velocity diffusion of ions arising due to overlapping of $\omega -\varOmega _i=n\varOmega _b$
resonances, where $\varOmega _b$
is the bounce frequency and $n$
is some integer value (Chernoshtanov Reference Chernoshtanov2016, Reference Chernoshtanov2018). Such diffusion can scatter ions away from the vicinity of the loss cone in relatively short times (approximately tens of bounce periods), which are small in comparison with the relaxation time of the fast ion distribution due to Coulomb collisions. In addition, the AIC instability stays coherent over time intervals exceeding 1 ms. These observations correspond to the model in which AICI is excited by one group of ions and leads to anomalous losses of another group of ions (Chernoshtanov Reference Chernoshtanov2016, Reference Chernoshtanov2018) in contrast with the DCLC instability, which is excited by ions near the loss cone and scatters those same ions; after the ions are lost, the DCLC burst decays. However, we cannot be sure that additional energy losses due to AICI stop completely after a short period because the time dependence of losses under the same conditions but without the instability is unknown, as these conditions inevitably lead to the instability forming; the same is true for the case of the DCLC instability.
Figure 13. Total power (black) and ion current (red) measured by the absorber diagnostics synchronised with magnetic probe signals (rescaled for visibility): (a) DCLC-type instability; (b) AIC instability.
4. Conclusion
We conducted an experimental study of kinetic plasma instabilities in the GDT, a magnetic mirror device, with target plasmas composed of various mixtures of hydrogen and deuterium (as well as pure hydrogen and deuterium plasmas) with deuterium beam injection. In these experiments a number of IC instabilities were observed; of most interest to us were the instabilities with frequencies of 2.2 MHz and 4.5 MHz, which we determined to be drift cyclotron loss cone instabilities. Although other high-frequency (25–80 MHz) and AIC instabilities were also present, they were not investigated in depth because the former seemed non-crucial for fast ion confinement and the latter had been investigated in detail in previous studies.
The instabilities at frequencies of 2.2 MHz and 4.5 MHz were observed in plasma targets with the linear electron density in the GDT centre ranging from $2.2\times 10^{14}$
to $3.6\times 10^{14}\ {\rm cm}^{-2}$
. These frequencies were close to the deuterium IC frequency in the GDT centre and in the fast ions turning points, respectively. The instabilities led to fast ion scattering, which was observed as drops in diamagnetic flux signals near turning points and increases in longitudinal energy and particle losses. With the use of plasma absorber diagnostics energies of the lost ions were estimated to be ranging from 4 to 16 keV.
Starting with a pure hydrogen target plasma, we gradually replaced hydrogen in the target mixture by deuterium. The properties of the 2.2 MHz instability did not change (the same properties of the 4.5 MHz instability could not be determined); however, when the deuterium content in the target reached a value between 54 % and 65 % the instability was suppressed entirely. As the deuterium percentage exceeded 65 %, the instabilities were only observed at 4.5 MHz, which presumably originated in the turning points as it could not be the result of nonlinear coupling of the absent 2.2 MHz instability. Increasing the plasma density to approximately $5\times 10^{14}\ {\rm cm}^{-2}$
resulted in complete suppression of the instability in deuterium targets and reduced the instability impact on diamagnetic fluxes in hydrogen targets.
We attributed the 2.2 MHz instability to be a DCLC type because of its several features:
(i) the instability frequencies were multiples of the IC frequency;
(ii) addition of warm ions led to the instability being suppressed, which was most effective when the warm ions were of the same isotope as the fast ones;
(iii) the instability was small scale, i.e. its normalised azimuthal wavenumbers mostly ranged from 2 to 5;
(iv) the unstable wave propagated in the ion diamagnetic drift direction.
Despite this, we currently cannot experimentally confirm its flute-like nature. An aggregate of the data obtained for the 2.2 MHz instability allowed us to establish that it scattered ions located near the loss cone in the velocity space. This is in agreement with the fact that the instability developed in incoherent bursts and the estimation that the scattered ions performed a few bounce oscillations before being lost. Although the instability at 4.5 MHz behaved in the same way as the one at 2.2 MHz in relation to warm ion addition and the fast ion scattering, an uncertainty regarding its azimuthal structure did not allow to definitively identify it as a DCLC instability.
In view of this, we propose the following interpretation of the obtained results. A certain quantity of warm ions is needed to prevent the DCLC instability and the suppression is the most effective when the frequencies of the IC harmonics for warm and fast ions coincide. With oblique injection of the neutral beams the fast ion density profile has two maxima near the turning points, so a potential well for the warm ions forms between them (if $n_{{\rm fast}} \gtrsim n_{{\rm warm}}$
). Thereby, the conditions required for the DCLC instability suppression occur in the trap centre vicinity first and foremost. Hydrogen plasma targets cannot suppress the instability at the first deuterium IC harmonic in the GDT centre, so the instability at 2.2 MHz is observed. The same situation remains in the case of mixed targets until the deuterium percentage of the target plasma exceeded 65 %. At the same time the amount of warm deuterium turns out to be insufficient for stabilising DCLC in the turning points regions, so the 4.5 MHz instability can appear even in plasmas composed entirely of deuterium and can only be suppressed by increasing the target density accordingly.
The evolution of the DCLC instability is a complex process, since it depends on several interrelated factors such as the degree to which the loss cone is filled with warm plasma, the isotopic composition of the plasma, the radial distributions of warm and fast ion densities as well as the axial distribution of the fast ion density. Therefore, the offered interpretation requires validation by numerical modelling of the fast ion distribution function while accounting for experimental conditions. Nonetheless, our results show that DCLC instability can be suppressed in the GDT by using two-species target plasmas, which improves the prospects of a GDT-based fusion neutron source.
Acknowledgements
Editor Cary Forest thanks the referees for their advice in evaluating this article.
Funding
The GDT maintenance, experiment conduction and data acquisition were supported by the Ministry of Science and Higher Education of the Russian Federation. Sections of the study concerning the probe data processing, determining DCLC instability properties and DCLC suppression conditions analysis were supported by Russian Science Foundation (Grant No. 19-72-20139).
Declaration of interests
The authors report no conflict of interest.
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