Laboratory study of kinetic instabilities in a nonequilibrium mirror-confined plasma

Resonance interaction of electromagnetic waves and charged particles plays an important role in the dynamics of magnetoactive plasma confined in space and laboratory magnetic traps. One of the most interesting manifestations of such interaction is the generation of powerful electromagnetic radiation and fluxes of energetic charged particles from the magnetic traps as a result of the kinetic plasma instabilities. Kinetic instabilities of nonequilibrium plasma are caused by the presence of positive gradients in the resonant particles velocity distribution, whose formation is universal for both space and laboratory plasma. In space magnetic traps (magnetosphere of Earth and planets, solar coronal loops, magnetosphere of stars) the sources of free energy stored in particles are formed due to different acceleration mechanisms (betatron acceleration, acceleration as a result of plasma-wave turbulence and magnetic reconnection). Under laboratory conditions energetic particles with an anisotropic velocity distribution can be formed due to the features of plasma heating, when the energy of the external source is embedded in a specific group of momenta of the particles (cyclotron heating, magnetic compression, beam heating). Another unifying feature is that the particles trapped in the adiabatic magnetic traps are characterized by a predominance of transversal velocities (relative to the direction of the magnetic field) to the longitudinal ones, which is expressed in the presence of the loss-cone particle velocity distribution. As a rule, plasma confined in the magnetic traps consists of at least two components, one of which, more dense and cold, determines the dispersion properties of the waves, and the second, a small group of energetic particles with an anisotropic velocity distribution, is responsible for the gain (absorption) of the waves. The interaction of high-frequency waves with resonant electrons leads to the diffusion of energetic electrons in velocity space and eventually to their falling into the loss cone and precipitation from the trap [1].

Cyclotron instabilities of two-component nonequilibrium plasma in adiabatic magnetic traps are among the most common and studied in vivo. Most vivid manifestations are the generation of powerful electromagnetic emissions in magnetospheres of the Earth (VLF-ELF emissions [2], Auroral Kilometric Radiation [3,4]), planets (Jupiter, Saturn) [5] and stars [6], as well as precipitations of energetic particles affecting the dynamics of the Earth's radiation belts [7]. Research of cyclotron instabilities of nonequilibrium plasma in the laboratory are highly relevant in terms of modeling the physical mechanisms of instabilities in space plasma [8]. Cyclotron instability is an important channel for the loss of excess energy stored in the electrons [9,10], thereby limiting the achievement of peak plasma parameters. For example, in modern sources of multiply charged ions, particles precipitation due to instabilities significantly modifies the distribution function of the hot electrons, thus limiting the average ion charge [11,12]. Electromagnetic radiation due to cyclotron instability is of interest itself for the diagnostics of plasma parameters in controlled thermonuclear fusion experiments [13,14]. Studies of cyclotron instabilities in magnetically trapped laboratory plasmas have a long history but still remain topical, mostly with the advent of powerful sources of microwave radiation (especially, gyrotrons), which allow to sufficiently raise the level of energy input into the plasma thereby increasing the "energetics" of nonequilibrium resonant particles.

In this paper we report about kinetic instabilities of nonequilibrium plasma heated by powerful radiation of the gyrotron under the electron cyclotron resonance (ECR) conditions and confined in the mirror magnetic trap. Instabilities are manifested as generation of short pulses of electromagnetic radiation accompanied by precipitation of hot electrons from the magnetic trap. The focus of this work was given to the study of time-frequency characteristics of the electromagnetic radiation, which has been made possible only recently with the advent of methods for measuring the electromagnetic field with high temporal resolution. Experimental data on the dynamic spectra of five different types of kinetic instabilities in various conditions of ECR discharge plasma are firstly presented.

The experiments were conducted in the plasma of ECR discharge sustained by powerful millimeter-wave gyrotron radiation (frequency 37.5 GHz, power up to 80 kW, pulse duration up to 1 ms) under the ECR conditions in the magnetic mirror trap. The schematic view of the setup is shown in fig. 1. Microwave radiation travelling through the input teflon window and the matching device is focused in the heating region of the discharge chamber. The axially symmetric discharge chamber is placed in a mirror magnetic trap of length 20 cm, produced by pulsed coils which allow to obtain a maximum magnetic field strength of 4.3 T and the mirror ratio is about 5. Plasma is created and heated under ECR conditions at the first cyclotron harmonic. The ECR zone is situated between the magnetic mirror point and the center of the discharge chamber and corresponds to a magnetic field strength of 1.34 T. Ambient pressure of a neutral nitrogen gas is about $10^{-6}\ \text{torr}$ and can reach values up to $10^{-3}\ \text{torr}$ during experimental shot.

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In the dynamics of pulsed ECR discharge three distinguishing stages with different plasma parameters can be defined. In the first stage, with a duration of about $100\ \mu \text{s}$, plasma density is small and microwave energy absorbed in a plasma volume is enough to heat electrons up to relativistic energies [15]. In the second stage, until the end of microwave pulse of gyrotron, plasma density is higher than during the first stage by more than two orders of magnitude. At that period of time plasma is two-componental containing a cold dense component $(N_c,\ T_c)$ with an isotropic velocity distribution, and a less dense component of hot electrons $(N_h,\ T_h)$ with anisotropic distribution function (with a predominance of the transversal to the magnetic field momentum as compared to the longitudinal one). Parameters of the plasma in a developed discharge are $N_c \sim 10^{13}\ \text{cm}^{-3}$, $N_h \sim 10^{11}\ \text{cm}^{-3}$, $T_c \sim 300\ \text{eV}$, $T_h \sim 100\ \text{keV}$ [16]. The third stage starts right after the microwave pulse termination in a decaying plasma, when the density of the cold fraction decreases rapidly, while the hot electrons with an anisotropic velocity distribution function are confined in the magnetic trap much longer. Therefore, starting from a certain time, the density of the hot component can become equal to or even higher than the density of the cold component [17].

In the experiments we studied the dynamic spectrum and the intensity of stimulated electromagnetic radiation from the plasma with the use of a broadband horn antenna with a uniform bandwidth in the range from 2 to 20 GHz and the oscilloscope Tektronix MSO 72004C (analog bandwidth 20 GHz, time resolution 10 ps). Using a pin-diode detector we measured precipitations of energetic $(>10\ \text{keV})$ electrons from the trap ends with time resolution of about 1 ns. As compared to previous studies [10,18] this experiment allowed us to observe the whole picture of instabilities and to distinguish details of the radiation spectrum.

Three stages of pulsed ECR discharge offer the opportunity to simultaneously study wave-particles interactions for essentially different plasma parameters: the initial stage, when the density of hot (relativistic) electrons exceeds the density of cold electrons $(N_h \gg N_c)$, the developed discharge $(N_h \ll N_c)$, and decaying plasma $(N_h \approx N_c)$ after the gyrotron switch-off. On each stage we detected a series of quasi-periodic broadband pulses of electromagnetic radiation (typical pulse duration is a few microseconds) and related precipitations of energetic electrons, which are caused by cyclotron instability of different electromagnetic modes or by generation of plasma waves. A typical dynamic spectrum of the electric field oscillations in the excited wave is shown in fig. 2. The frequency bands of the observed radiation can be compared in the dynamic spectrum with the help of the electron cyclotron frequency f_ce0 in the center of the magnetic trap which is changing in time.

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At the start-up phase of the ECR discharge (area No. 1 in fig. 2) we detected microwave radiation in a direction perpendicular to the ambient magnetic field at frequencies slightly higher than the electron cyclotron frequency f_ce0. We explored two significantly different regimes of instabilities: periodic wave packets with fast raising frequency smoothly transforms into the continuous spectrum of radiation [19]. In both cases the bandwidth of the registered radiation is several GHz. Also the second harmonic of this signal was observed (see fig. 2).

During the developed discharge phase we registered microwave radiation in a direction longitudinal to the ambient magnetic field at frequencies about $f_{ce0}/2$ (area No. 2 in fig. 2) and corresponding precipitations of hot electrons from the trap ends. At the same time the other type of radiation at frequencies between f_ce0 and 2f_ce0 was observed (area No. 3 in fig. 2). The dynamic spectrum of the latter has a sharp upper boundary which varies in proportion to the frequency of 2f_ce0.

With a delay of about $30\ \mu \text{s}$ after ECR heating switch-off the instability was detected as a long series of periodic broadband pulses (up to hundreds of bursts) of electromagnetic radiation at frequencies near 2f_ce0 (area No. 4 in fig. 2). The electromagnetic energy released during this particular phase is more than a half of the total energy emitted during the entire decay phase. Next to periodic series with a delay of several hundreds of microseconds quasi-periodic broadband bursts of radiation with fast falling frequency with much higher period were observed (area No. 5 in fig. 2).

The experimental shot depicted in fig. 2 is rather general and shows us a typical situation when all types of instabilities are observed simultaneously. But particular details of electromagnetic spectrum may vary from one shot to another while all parameters of the setup are the same. Therefore, we chose the most typical implementation of these types of instabilities which are shown in fig. 3 in the same order as in fig. 2. In fig. 3 panels (b) and (c) correspond to the same experimental shot as in fig. 2. Dynamic spectra shown in the other panels correspond to the other experimental shots that differ only in the magnetic field strength. Panels (a) and (e) correspond to instabilities in rarefied plasma, panels from (b) to (d) correspond to instabilities in dense plasma.

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In fig. 3(a) the regime of periodic wave packets with fast raising frequency is shown. It is seen from the spectrogram that the lower spectrum boundary of radiation is greater than f_ce0 on the value of about 0.01f_ce0. The frequency change rate in a single wave packet is more than 5 GHz per μs.

In fig. 3(b) quasi-periodic bursts of radiation of dense plasma near $1/2f_{ce0}$ are shown. Every radiation pulse is strongly correlated with energetic electrons precipitations measured by detectors. The duration of pulses is about $1\ \mu \text{s}$. The distinctive feature of this type of instability is the presence of the selected frequencies (more than ten) in the spectrum, which are arranged equidistantly relatively to each other. These frequencies of spectral components are slightly changing in time while the distance between them is constant. In fig. 3(b) the distance between spectral components is $\Delta f \approx 150\ \text{MHz}\ (\Delta f/f\approx 0.05)$ and the spectral width of a single component is $\Delta f_{\textit{single}}\approx 20\ \text{MHz}$ ($\Delta f_{\textit{single}}/f\approx 7\times 10^{-3}$, $\Delta f_{\textit{single}}/\Delta f \approx 0.1$).

In fig. 3(c) the high-frequency radiation of dense plasma is shown. The upper cut-off frequency $f_{\ast}$ is clearly observed and $(2f_{ce0}-f_{\ast}) \approx 0.1f_{ce0}$. The spectrum of radiation is rather complex but we can distinguish single wave packets with decreasing frequency.

In fig. 3(d) periodic bursts of radiation just after ECR heating switch-off are shown. In this period of time the density of a cold plasma is decreasing in time due to plasma decay but is still greater than electron gyrofrequency. That is why we can consider that the plasma is dense. Two different types of radiation can be noted in the spectrogram. The first one, in contrast to previous types of instability, is a periodic broadband radiation without frequency sweep in a single burst. The spectral width of a broadband radiation is $\Delta f/f\approx 0.1$. The period of oscillations is about 200 ns and a single burst duration is about 50 ns. The second type of radiation is observed simultaneously with the previous one. These are bursts of a narrowband radiation $(\Delta f/f\approx 4\times 10^{-3})$ at frequencies below the broadband radiation frequencies with a duration about $1\ \mu \text{s}$. The frequency of a narrowband radiation is changing in time in proportion to 2f_ce0.

The last type of instability is shown in fig. 3(e). Plasma emission during decay stage is a sequence of periodic bursts with decreasing frequency within a single burst. The duration of every burst is about $1\ \mu \text{s}$ and the period varies from 30 to $50\ \mu \text{s}$. Almost every burst starts with a narrowband emission at frequency slowly decreasing in time. Then during a very short period of time (about 100 ns) the frequency of radiation rapidly drops with a speed of about 5 GHz per μs. The moment of fast frequency drop is correlated with energetic electrons precipitation from the trap ends. The frequency of bursts in a sequence is below 2f_ce0 and also proportional to it.

As seen from experimental data the general picture of plasma emission due to kinetic instabilities is rather complex. But all peculiarities in the dynamic spectrum of radiation are strongly correlated with changing in time electron cyclotron frequency $f_{ce0}= \omega_{ce0}/2\pi$ in the trap center. This means that the observed radiation is more likely excited under electron cyclotron conditions and the region of effective wave-particle interaction is situated in the center of the magnetic trap, where the magnetic field is the most homogeneous. Location of the most effective interaction area in the region of the most homogeneous magnetic field was discussed earlier in relation to excitation of electromagnetic waves as a result of cyclotron instability of the Earth's radiation belt electrons [2]. In the region of homogeneous magnetic field the excited wave has a maximum growth rate averaged over the amplification path.

Interpretation of the observed data is based on the concept of the dense background plasma acting as a trigger of a cyclotron maser instability that released the energy of hot electron component typical of ECR discharge [20].

While the dense plasma is absent during the initial phase of discharge the radiation is observed at frequencies exceeding the electron gyrofrequency at the trap center (type 1 in fig. 2). Therefore, it is more naturally explained by the excitation of the loss-cone instability of a fast extraordinary (X-mode) wave, which is typical of the mirror traps and propagates oblique to the magnetic field [19]. Dynamic spectrum of plasma radiation at the initial stage of the discharge can be used to estimate energies of resonant electrons, as was done in [19].

The most favorable conditions of excitation of the loss-cone instabilities are realized in the case where the resonance curve is entirely in the loss region in the velocity space and touches the loss cone [21]. In the approximation of a spatially uniform plasma these conditions are determined as

$$\begin{equation} \omega-k_{\parallel}v_{\parallel}-\omega_{ce}/\gamma=0,\qquad v_{\perp}/v=1/\sqrt{R}, \end{equation} \tag{ 1 }$$

where $k_{\parallel}$ is the wave vector projection onto the external magnetic field direction, $v_{\parallel}$ and $v_{\perp}$ are the components of electron velocity v along and across the magnetic field directions, $\gamma=(1-v^2/c^2)^{-1/2}$ is the relativistic factor and R is the mirror ratio of the trap. Using conditions specified in eq. (1) we can obtain the relation between the radiation frequency ω and the kinetic energy K of the radiating electron, which is determined at the contact point of the resonant curve and the loss cone in the velocity space:

$$\begin{equation} K=mc^2(\gamma-1)=mc^2(\omega/\omega_{ce}-1). \end{equation} \tag{ 2 }$$

Thus, the presence of an upper boundary of the radiation frequency $\omega_{\textit{max}}$ allows us to estimate the maximum energy of the hot resonant electrons. For the case in fig. 3(a) $\omega_{\textit{max}}/\omega_{ce}\approx 1.2$, which indicates that electrons are accelerated up to 100 keV at the initial discharge stage. In the individual experiments the ratio $\omega_{\textit{max}}/\omega_{ce}$ can have values up to 1.6, which corresponds to the energies up to 300 keV. This agrees well with the previously published data on the measurement of the electron energy distribution [15].

At a large density of the background plasma during the second discharge stage, cyclotron instabilities of extraordinary modes with quasi-perpendicular propagation to the magnetic field are suppressed, because their dispersive properties are strongly modified by the background plasma. Emission of dense plasma at frequencies about $f_{ce0}/2$ (type 2 in fig. 2) is more likely related to the whistler mode instability. The instability of whistler mode waves at this laboratory setup was firstly observed in [18] where estimations of plasma radiation spectrum were obtained using microwave detectors. In the present work we can derive dynamic spectrum and observe its fine structure (see fig. 3(b)) which was not possible before. However discussing the fine structure of whistler waves spectrum at this experiment is out of the scope of the paper.

At the same time the observed radiation at higher frequencies with the sharp upper spectral boundary (type 3 in fig. 2) apparently can be related to the excitation of plasma waves under upper hybrid resonance. Plasma waves propagate perpendicular to the ambient magnetic field and then transform to electromagnetic waves (e.g. slow extraordinary waves) in rarefied plasma at boundary regions of the trap.

With the use of mirror-confined plasma produced by the ECR discharge we provide the possibility to study plasma instabilities under double plasma resonance condition in the laboratory [22]. In the experiment such conditions are fulfilled just after ECR heating switch-off, i.e. at the very beginning of a dense plasma decay phase (type 4 in fig. 2). During the plasma decay the upper hybrid resonance frequency $\omega_{uh}=(\omega_{pe}^2+\omega_{ce}^2)^{1/2}$ is also decreasing and at the moment of instability development it is equal to the second harmonic of gyrofrequency. In the case of double plasma resonance condition when the frequency of the upper hybrid resonance coincides with one of the electron gyrofrequency harmonics the instability growth rate of plasma waves is greatly increased. This leads to the appearance of bright narrow-band radio emission near the harmonics of the electron gyrofrequency – the so-called zebra patterns [23,24]. In the experiment we observe double plasma resonance only for one cyclotron harmonic which is limited by the bandwidth of the used oscilloscope. In this case the pulsing mode of plasma waves generation and synchronous precipitations of fast electrons from the trap may be related either to the competition of instability and induced scattering in the generation of plasma waves, or to the excitation of fast magnetosonic waves in the magnetic trap. Since under experimental conditions high growth rates of plasma waves are implemented, the first cause of the pulsations seems more favorable.

It should be noted that the possible manifestations of a double plasma resonance effect are not rare in astrophysical plasmas. The phenomenon of zebra pattern is observed in the solar corona (type IV bursts) [24], in the decametric radiation of Jupiter [25], in VLF radiation of the Earth's magnetosphere [26] and even in the radio emissions of pulsars [27]. In connection with the above, verification of the effect of double plasma resonance in a laboratory experiments is a very relevant task.

After the gyrotron is switched off (type 5 in fig. 2), the burst activity is also attributed to loss-cone instabilities of the extraordinary waves propagating in perpendicular direction to the magnetic field [17,28,29]. A relatively new nonlinear regime of electron cyclotron instability aimed at the explanation of complex temporal patterns of detected electromagnetic radiation observed in a decaying plasma was discussed in [30]. This regime is characterized by self-modulation of a maser due to the interference of two counter-propagating unstable waves resulting in spatial modulation of amplification. The resulted maser dynamics shows rather complex behaviour similar to what was detected experimentally (single spikes may join in bursts, the interval between spikes may become irregular, generation may be switched to a stochastic regime). The same approach was used to explain the transition between two regimes of instability development when periodic wave packets with fast raising frequency smoothly transforms into the continuous spectrum of radiation at the start-up phase of the ECR discharge [19].

An important advantage of nonequilibrium ECR discharge plasma is an opportunity to recreate different conditions for excitation and amplification of waves in plasma in a single discharge pulse has been demonstrated in the paper. Various dynamic spectra of electromagnetic radiation related to at least five types of kinetic instabilities have been observed for the first time. Generation of pulsed electromagnetic radiation accompanied by hot electrons precipitation from the trap has much in common with similar processes occurring in the magnetosphere of Earth, planets, and solar coronal loops. The paper may be of interest in the context of a laboratory modeling of nonstationary processes of wave-particle interactions in nonequilibrium space plasma, since there are a lot of open questions about the origin of some types of emissions in space cyclotron masers [31], especially in mechanisms of fine spectral structure. Also recent studies about the electron cyclotron instabilities impact on the quality of the ion beams in the ECR ion sources show the importance of the present work for practical applications [11,12].

The work has been supported by RFBR (grants No. 13-02-00951, 14-02-31521) and Russian Academy of Sciences program OFN-15. The work of MV was also supported by the Presidential Council for Young Russian Scientist Support (grant No. SP-4857.2013.3) and the Dynasty foundation.