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Observation of quasi-periodic frequency sweeping in electron cy-

clotron emission of nonequilibrium mirror-confined plasma

M. E. Viktorov(a), A. G. Shalashov, D. A. Mansfeld and S. V. Golubev

Institute of Applied Physics of Russian Academy of Sciences – 46 Ulyanov Str., 603950, Nizhny Novgorod, Russia

PACS 52.35.-g – Waves, oscillations, and instabilities in plasmas and intense beamsPACS 52.72.+v – Laboratory studies of space- and astrophysical-plasma processes

Abstract – Chirping frequency patterns have been observed in the electron cyclotron emissionfrom strongly nonequilibrium plasma confined in a table-top mirror magnetic trap. Such patternsare typical for the formation of nonlinear phase space structures in a proximity of the wave-particleresonances of a kinetically unstable plasma, also known as the “holes and clumps” mechanism.Our data provides the first experimental evidence for acting of this mechanism in the electroncyclotron frequency domain.

Introduction. – Resonant interaction of electromag-netic waves and charged particles plays an important rolein the dynamics of magnetoactive plasma confined in spaceand laboratory magnetic traps. One of the most intrigu-ing manifestations of such interaction is emission of broad-band radiation with regular variations of dynamical spec-tra, e.g. quasi-periodic bursts with a frequency sweeping,resulting from the development of kinetic plasma insta-bilities. Such events are common features of experimen-tal plasmas. Kinetic instabilities are caused by the pres-ence of positive gradients in the velocity distribution ofresonant particles, whose formation is universal for bothspace and laboratory plasma. In space magnetic traps,the sources of free energy are formed due to different ac-celeration mechanisms of particles, such as betatron accel-eration, plasma-wave turbulence and magnetic reconnec-tion. Under laboratory conditions, the energetic particleswith an anisotropic velocity distribution can be formeddue to the features of plasma heating, when the energyof the external source is embedded in a specific region ofthe phase space, such as provided by resonant cyclotronheating or neutral beam injection in magnetic fusion ex-periment. Spatial gradients can also result in instabilitiesof waves for which the diffusion in real space is coupled tothe diffusion in velocity space due to conservation of invari-ants of particle motion in inhomogeneous systems energy;this type of instability is exploited in the so-called alphachanneling proposals [1, 2]. As a rule, plasma confined inlaboratory traps consists of at least two components, one

(a)E-mail: mikhail.viktorov@appl.sci-nnov.ru

of which, more dense and colder, determines the disper-sion properties of the unstable waves, and the second smallgroup of energetic particles with an essentially anisotropicvelocity distribution which is responsible for the growth ofthe waves.

One common description for the self-consistent evolu-tion of particles and waves is the quasilinear theory, a per-turbative approach that involves many overlapped wave-particle resonances as a basis for a diffusive particle trans-port in phase space1 [3,4]. Within this approach, which ismost developed for the whistler and Alfven wave instabili-ties in space [8], the frequency sweeping may be explainedas a result of relatively slow modification of the averagedistribution function of the resonant particles. Althoughthe quasilinear theory is in principle able to describe fastevents, such as switching from kinetic to hydrodynamic in-stability in the inner magnetosphere of the Earth [9–11],it faces essential difficulties in many cases involving fasttransients.

An alternative universal physical mechanism is basedon formation of nonlinear phase space structures in theproximity of the wave-particle resonances of a kineticallyunstable bulk plasma mode. In collisionless limit, suchregimes were proposed as a possible mechanism of genera-tion of narrowband chorus emissions in the Earth’s magne-

1Resonant interaction between monochromatic wave and parti-cles in inhomogeneous magnetic field may also result in quasi-lineardiffusion in momentum space, that happens because of the stochas-tic phase of particles during repeated passing trough the resonanceregion, see e.g. [5]. Such concept of quasi-linear diffusion is widelyused related to a high-frequency heating of fusion plasmas [6, 7].

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M. Viktorov et al.

tosphere [12–14]. Evolution of similar phase space struc-tures was studied in [15–17] for the case of essential dissi-pation, which is more relevant to our situation. Here thewave resonances do not overlap and the global transportin phase space is suppressed. As the mode grows, mostof the particles respond adiabatically to the wave, andonly a small group of resonant particles mix and cause lo-cal flattening of the distribution function in phase spacewithin or near the separatrices formed by the waves. How-ever, when linear dissipation from a background plasmais present, the saturated plateau state becomes unstableand the mode tends to grow explosively. That results inthe formation and subsequent evolution of long-living, ascompared to the linear growth, structures in the parti-cle distribution, so called holes (a depletion of particles)and clumps (an excess of particles). These structuresrepresent nonlinear waves of so-called Bernstein–Greene–Kruskal type [18] whose frequencies are slightly up- anddown-shifted with respect to that of the initial instabil-ity. Their subsequent convective motion in phase space issynchronized to the change in wave frequency, thus lead-ing to complex chirping patterns in dynamical spectra ofunstable waves, as discussed in more details in [19–21].This mechanism is frequently referred as Berk–Breizmanmodel. Formation of holes and clumps is commonly con-sidered as near-threshold phenomena. However recently ithas been realized that the same processes may occur farfrom the instability threshold as well [22]. The underlyingphysics is that holes and clumps develop from negativeenergy waves.Over the years, the hole and clump mechanism has been

extensively examined to study and interpret the Alfvenwave turbulence driven by high-energy beam ions or alphaparticles in toroidal magnetic traps [17, 19–21, 23–28]; aswell as some other MHD processes, such as fishbone [17]and ion-acoustic [29] instabilities, have been considered.In this letter, we report the first laboratory observa-

tions of potentially the same mechanism acting in a muchhigher frequency domain. We study the electron cyclotron(EC) kinetic instabilities of plasma with fast electrons sus-tained by high-power microwave radiation under electroncyclotron resonance (ECR) conditions and confined in themirror magnetic trap. On a decaying phase of a dis-charge after the microwave heating is switched-off, we findthe chirping frequency patterns in the plasma emission inthe EC frequency range that are very similar to thosepredicted by Berk–Breizman model. The focus of thiswork was devoted to the study of peculiar time-frequencycharacteristics of the electromagnetic radiation, which hasbeen made possible only recently with the advent of meth-ods for measuring the electromagnetic field with high tem-poral resolution.It should be noted that research of cyclotron instabili-

ties of nonequilibrium plasma in the laboratory are highlyrelevant in terms of modeling the physical mechanismsof instabilities in space plasmas [30, 31]. Using ECR dis-charge sustained by high-power microwave radiation (gen-

erated with modern gyrotrons) in a magnetic trap allowsto increase the energy deposited into the nonequilibriumplasma component as compared to early experiments. An-other important advantage of modern laboratory study ofECR discharge is an opportunity to recreate very differ-ent conditions for excitation and amplification of waves inthe same setup. As a result, generation of pulsed electro-magnetic radiation in a laboratory has much in commonwith similar processes occurring in the magnetosphere ofEarth [8,32], other planets [33], solar coronal loops [34–36],stars [37], etc.

Experimental conditions and results. – The ex-periments were conducted in the plasma of ECR dischargesustained by gyrotron radiation (frequency 37.5GHz,power up to 80 kW, pulse duration up to 1ms) in thesimple axially symmetric open magnetic trap. Microwaveradiation traveling through the input teflon window andthe matching device is focused in the heating region of thedischarge chamber. The radiation intensity in the focalplane is about 10 kW/cm2 and the average power densityis 100W/cm3. Discharge chamber is a tube with innerdiameter of 38mm which is widened in center part by thetube with inner diameter of 72mm and 50mm length. Adischarge chamber is placed in the mirror magnetic trapof the length 225mm, produced by pulsed coils with maxi-mum magnetic field strength of 4.3T, mirror ratio is about5, pulse duration 7ms. Plasma is created and supportedunder ECR conditions at the fundamental cyclotron har-monic corresponded to the magnetic field strength 1.34T.The resonance surface is situated between the magneticmirror and the center of the discharge chamber. Ambi-ent pressure of a neutral gas (nitrogen or argon) is about10−6Torr, however it increases up to 10−4−10−3Torr dur-ing ECR discharge. Further details on the setup may befound in [38].

In the experiments we studied the dynamic spectrumand the intensity of stimulated electromagnetic radiationfrom the plasma with the use of a broadband horn an-tenna with a uniform bandwidth in the range from 2 to20GHz and the high-performance oscilloscope KeysightDSA-Z 594A (analog bandwidth 59GHz, sampling rate160GSample/s). To cover all stages of ECR discharge thelength of the recorded waveforms was set to 5ms whichcorresponds to 8×108 data points per channel for a singleexperimental shot. The dynamic spectra were calculatedfrom the recorded data by short-time Fourier transformwindowed with a Hamming window. Simultaneously, wemeasured precipitations of energetic electrons (> 10 keV)from the trap ends using a pin-diode detector with timeresolution about 1 ns. As compared to the early studies[38–42], this experiment allowed us to observe the wholepicture of instabilities and distinguish details of the radia-tion spectrum. Similar technique was used in our previousstudies [36, 43, 44].

Operation of an external microwave heating definesthree time slots within every experimental shot: the ECR

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Observation of quasi-periodic frequency sweeping in electron cyclotron emission

Fig. 1: Electric field oscillations in the microwave emission during the nitrogen plasma decay stage (a) and correspondingdynamic spectrum (b)-(e). Panels (c)-(e) shows the same spectrum in zoom areas. The white line on the spectrograms showstemporal variation of the on-axis electron cyclotron frequency fce0 = ωce(zcenter)/2π at the magnetic trap center. The ECRheating is operating during 0 − 1ms (not shown). Horizontal dark blue lines in the specra are rational fractions of Nyquistfrequency and related to a digital noise.

breakdown and the plasma start-up, the developed dis-charge and the plasma decay stage after the microwavepump switch-off. In the start-up stage, with a dura-tion of about 100µs, plasma density is small and mi-crowave energy absorbed in a plasma volume is enoughto heat electrons up to relativistic energies [43]. In the de-veloped discharge stage, until the end of the microwavepulse, the plasma density is by more than two ordersof magnitude higher than during the first stage. Here,the two-component electron population is formed contain-ing the cold dense electrons with Maxwellian distribution(Nc ∼ 1013 cm−3, Tc ∼ 300 eV), and less dense compo-nent of hot electrons with anisotropic distribution function(Nh ∼ 1011 cm−3, mean energy Th ∼ 15 keV, although en-ergy tail is stretched up to 200−300 keV). The third stagestarts right after the microwave pulse termination in a de-caying plasma, when the temperature and density of thecold fraction decreases rapidly, while the hot electrons are

adiabatically confined in the magnetic trap much longer.Therefore, starting from a certain time, the density of thehot component can become comparable to or even higherthan the density of the cold component [38].

Different stages of pulsed ECR discharge offer the op-portunity to simultaneously study wave-particle inter-actions for essentially different plasma parameters [44].However, highly transient spectra with many repeatedfrequency sweeps have been observed only in a decayingplasma (Nh & Nc) and only with a pronounced delay afterthe ECR heating switch-off. Typical example of such eventin a nitrogen discharge is shown in fig. 1. Excitation ofsuch chirping wave packets is observed after the microwaveheating switch-off with a delay from 0.1 to 1ms and onlywhen ambient magnetic field is decreasing in time. Fre-quency of the observed microwave emission is always belowthe electron cyclotron frequency fce0 at the trap center.The instability development starts with the pulse of con-

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M. Viktorov et al.

Fig. 2: Single-sided amplitude spectrum of the waveform shownin fig. 1(a). Frequency bands 4.9 − 5.3GHz, 6.2 − 6.5GHz,6.9− 7.1GHz, 7.2− 7.3GHz and 7.5− 7.6GHz are highlightedby green lines.

tinuous broadband emission in the frequency range from7 to 8 GHz. After that initial pulse which lasts approx-imately 50µs, the spectrum of emission is significantlychanged. The microwave emission is observed only in afew narrow frequency bands 7.5− 7.6GHz, 7.2− 7.3GHz,6.9 − 7.1GHz, 6.2 − 6.5GHz and 4.9 − 5.3GHz, that areindependent of all experimental conditions. These fre-quency bands are clearly seen in a single-sided amplitudespectrum of the whole waveform shown in fig. 1(a), seefig. 2. Within each frequency band the dynamic spectrumrepresents a set of highly chirped narrowband radiationbursts with both increasing and decreasing frequencies.The bandwidth of this emission inside a single wave packetis about 2× 10−3fce0. The duration of the single narrow-band spike is about 10µs; the overall duration of a seriesof such pulses is up to 1ms depending on the experimentalconditions.An important feature of the band microwave emission is

that in most cases it is observed only when its frequencyis below the electron gyrofrequency fce0 at the trap cen-ter. When the magnetic field decreases such that fce0becomes lower than the corresponding frequency band,the microwave emission in this particular band is not de-tected anymore. Simultaneously, the microwave emissionat lower frequency bands may still be detected or eventriggered on in a new band.The fine structure of the emission spectrum in a particu-

lar frequency band is changing with the ambient magneticfield decrease. When fce0 is significantly higher than emis-sion frequency, we observe predominance of rising toneswith sweeping frequency that saturates at more or less con-stant level for all neighboring pulses. With the decrease offce0, falling tones appear in the spectrum. Finally, whenfce0 is close to the frequency of the microwave emission, anarrowband continuous signal without pronounced chirp-ing patterns is observed. Thus, the dynamic bandwidthof the emission is reduced with the decrease of differencebetween the gyrofrequency and the emission frequency.Most of the discussed experiments have been performed

Fig. 3: Electric field oscillations (a) and corresponding dynamicspectrum (b) for the microwave emission during the decay stageof argon discharge.

in nitrogen plasma. However, an essentially similar dy-namics was observed in argon plasma as well. Figure 3shows an example of the frequency sweeping in a decay-ing plasma of argon discharge. Although less representa-tive statistics is available for argon, it seems that chirpedemissions of the argon discharge are possible in a more nar-row parameter range than for the nitrogen plasma. Thisis related, eventually, to more slow decay of the back-ground plasma with argon ions as compared to the ni-trogen plasma (the latter is dominated with dissociativerecombination of molecular ions [38]).

Unlike all other types of instabilities observed earlier,in the studied case we do not detect the precipitation ofhot electrons from the trap that usually accompanies theelectromagnetic emission.

Discussion. – The reported dynamic spectra of theplasma emission look very similar to those predicted byBerk–Breizman model in lower frequency domains [19,22,27]. Before comparison to this model, let us summarizeshortly our previous knowledge on the kinetic instabilitiesin a decaying plasma at our setup.

Such instabilities were first observed as quasi-periodicseries of pulsed energetic electron precipitations that ap-peared with some delay after the switching off the heatingpower [38], later rough estimations of plasma EC emis-sion spectrum were obtained using microwave detectors[42]. The instability was interpreted as a result of reso-nant interaction at the fundamental cyclotron harmonicbetween the energetic electrons and the slow extraordi-nary waves propagating in a rarefied plasma across theexternal magnetic field at frequencies below the local gy-rofrequency [38]. The linear growth rate was estimatedas γL ∼ 5 × 107 s−1. The delay between the start of thedecay and triggering of the instability was explained by

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Observation of quasi-periodic frequency sweeping in electron cyclotron emission

a polarization depression effect of the background (moredense and cold) plasma. In an unbounded dense plasma,the slow extraordinary wave is nearly circular polarizedwith the rotation direction opposite to the electron gyra-tion direction, therefore, the wave-electron interaction isstrongly weakened. However, in a rarefied plasma satisfy-ing the condition

ω2pe ≪ ω2

ce − ω2 ∼ ω2ce ε/mc2 (1)

the extraordinary wave is polarized almost linearly, whichensures an effective interaction with resonant electrons.Here ω, ωpe, ωce are, correspondingly, the wave, electronLangmuir and cyclotron frequencies calculated at the trapcenter, and ε is the kinetic energy of a radiating elec-tron. Hence, the cyclotron instability growth rate in-creases with the decreasing background plasma density.With ck||/ωce ≪

√

ε/mc2, the interaction occurs at therelativistically downshifted electron gyrofrequency

ω ≈ ωce/(ε/mc2 + 1) . (2)

The sequences of pulsed bursts at the nonlinear insta-bility phase was explained in terms of a cyclotron masermodel with a fast decrease of electromagnetic energy losses[40]. It has been shown that, even in the absence of a con-tinuously acting source of nonequilibrium particles (inver-sion of the medium), the instability condition, γL > γdwith γd ≈ νei being the wave damping rate due to col-lisional absorption by the background plasma, is recov-ered after each individual burst due to monotonic de-crease of νei(t) during the plasma decay. Therefore, thedynamics of the background plasma is very essential forthe proposed model since decreasing wave losses in thebackground plasma actually pump the maser instability.Unfortunately, experimental measurements of the back-ground plasma parameters have been possible only for thesteady state ECR discharge and no reliable data are avail-able describing the decay phase. So we reconstruct theelectron density and temperature during the plasma de-cay basing on the particle and energy balance equationssolved with initial conditions corresponded to the mea-sured parameters of the ECR discharge [38, 41].In the present work we derive dynamic spectrum and

observe its fine structure, which was not possible before.In fig. 4 we show the typical evolution of plasma frequen-cies and collisional rates during the plasma decay, corre-sponded to case shown in fig. 1. One can see that insta-bility starts in t∗ ≈ 2 ms when condition (1) becomes full-filled, ωpe/ωce ∼ 0.5. At this moment, the electron tem-perature Te is about 0.1 eV, the background plasma den-sity is Ne ∼ 2× 1011 cm−3, and the characteristic growthrate is γL ≈ γd(t∗) ≈ 6.3× 107 s−1. Dynamic spectrum ofthe plasma emission can be used to estimate energies ofemitting electrons from the resonant condition (2). For thecase shown in fig. 1, maximum detuning ωce/ω ≈ 1.32 isobserved at the initial phase of the instability developmentjust after the first broadband pulse. This indicates that

1.0 1.5 2.0 2.5 3.0 3.50

2

4

6

8

10

12

14

Time HmsL

Freq

uenc

yHG

HzL

fpe

ýd

2 þ´103

fce0

Fig. 4: Evolution of the plasma frequencies fpe, fce, the back-ground collision rate γd during the plasma decay. The timeinterval where the frequency sweeps are observed is shown ingray. Used model is discussed in [38,41]; modeling is performedfor the following initial conditions relevant to the experimen-tal shot shown in fig. 1: Ne(0) = Ni(0) = 2 × 1013 cm−3,Nm(0) = N+

m (0) = 0, Te(0) = 300 eV. A constant flux ofpuffed gas is assumed, F = 1.5×1016 cm−3 s−1. Energy of thefast electrons is ε ≈ 165 keV.

electrons are accelerated up to 165 keV. In a consequentseries of repetitive chirped pulses the radiation frequencyapproaches the gyrofrequency, ωce/ω → 1, so the energyof resonant electrons is decreasing.

Isolated narrow frequency bands observed in the mi-crowave emission spectrum may be explained by excitationof individual modes of a bounded plasma in a dischargechamber. As a rough estimate supporting this idea, onemay consider first several vacuum modes of the cylindri-cal plasma chamber. Such waveguide has cut-off frequen-cies 5.08GHz, 6.54GHz, 6.80GHz, 7.07GHz, 7.32GHzetc, that are very close to the observed frequency bands(fig. 2). Of course, a more detailed analysis of excitationof a plasma-filled waveguide is needed for quantitative de-scription of a linear phase of the instability. A similarproblem was studied in [45].

Our previous attempt to explain the observed complexdynamics demonstrated by the cyclotron maser in a de-caying plasma discharge was based on the idea of self-modulation of a maser due to interference of two counter-propagating degenerate unstable waves resulting in spatialmodulation of amplification [41]. However this model cannot describe frequency sweeps resolved in the reported ex-periments.

The “holes and clumps” paradigm, as formulated byBerk and Breizman, seems to be the most suitable modelto explain our data. Following [16] one can isolate onespectral component and model it by a Bernstein–Greene–Kruskal wave to obtain the time evolution of one chirpingevent. Recall that γL is the linear growth rate in the ab-sence of fast particle collisions and external dissipation,and γd is the damping rate defined by collisional absorp-tion of waves by the background plasma. Then, in the col-

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M. Viktorov et al.

lisionless limit for resonant fast particles, a bounce averageof Maxwell–Vlasov kinetic equation yields the saturationlevel ωb and the frequency shift δω of an unstable modeas

δω ≈ 16√2

3√3π2

γL√γdt , ωb ≈

16

3π2γL , (3)

where the bounce frequency of particles that are deeplytrapped in the wave potential, ωb, is used as a measure ofthe electric field amplitude. These expressions are validfor

|dωb/dt| ≪ |dδω/dt| ≪ ω2b ∼ γ2

L, (4)

i.e. in a regime where perturbation of the passing particledistribution is negligible. Although these analytical re-sults were originally obtained for purely electrostatic andone-dimensional bump-on-tail instability, it gives reason-able qualitative estimates for more complex electromag-netic problems, see e.g. [25]. Similar results were obtainedbefore for the particular case of cyclotron instability ofwhistler waves in Earth magnetosphere: ωb ≈ (32/3π)γLin [46] and ω2

b ≈ 2π dδω/dt in [12].Following this logic and understanding that more rigor-

ous analysis should follow, we apply eqs. (3) to our data.Fitting the measured frequency sweeps δω =

√A t by con-

stant A, one may estimate the upper boundary for thedissipation γd, or, equivalently, the lower boundary forthe linear growth rate γL compatible with Berk–Briezmanmodel. Indeed, the first equation (3) implies that γ2

Lγd ≈5A, and instability condition requires γL > γd. Evidently,the above mentioned boundaries correspond to the insta-bility threshold, γd = γL ≈ 3

√5A. In the experiments we

obtain typical values A = (0.4− 2)× 1021 s−3, what corre-sponds γL ≈ (1− 2.5)× 107 s−1. For example, fig. 5 showsthe evolution of the growth rate obtained by this tech-nique to the spectrogram (c) in fig. 1. Here each pointcorresponds to a separate chirping burst. Note that thissimple estimate, that neglects all geometrical factors re-lated to our particular modes, results in slightly underes-timated instability growth rates, but still is in a relativelygood agreement with our previous calculations.For the slow extraordinary waves propagating trans-

verse to the external magnetic field in a rarefied plasmathe bounce frequency may be approximated as [46]

ω2b ≈ v⊥

c

E∼

B0

ω2ce

γ2, (5)

here E∼ is a wave electric field, B0 is the external magneticfield strength, and v⊥ is the transverse electron velocity.Then from the second expression in (3) one can estimatethe characteristic level of saturated electric field as E∼ ≈2− 3V/cm for 60keV electrons.The above preliminary conclusions need further verifi-

cation by full non-linear solution of a kinetic equation forthe resonant particles, the work to be published separately.Note that the asymmetry of the ascending and descendingfrequency sweeps suggests an important role of the drag

2.04 2.06 2.08 2.1 2.12Time (ms)

1

1.5

2

2.5

3

γL (

s-1)

×107

Fig. 5: The lower boundary for the linear growth rate γL com-patible with Berk–Briezman model for the high frequency partof the dynamic spectrum shown in fig. 1(c).

force (either collisional, or turbulent) acting on the reso-nant particles [19,21,24,27]. Therefore, a realistic collisionoperator with the “drag” and “diffusion” parts should betaken into account in the kinetic modeling.

Summary. – In this paper the kinetic EC instabil-ities of two-component plasma sustained by high-powermicrowave radiation under ECR conditions and confinedin the mirror magnetic trap were studied. The chirpingfrequency patterns in the plasma EC emission, which arevery similar to those predicted by Berk–Breizman model,were observed during plasma decay stage, characterized byhigh relative density of the fast electrons. The fine struc-ture of dynamic spectra of EC emission was investigatedwith high temporal resolution. The analysis of the experi-mental data allowed to estimate energies of emitting elec-trons (165keV) and the range of instability growth rates(up to 6× 107 s−1).

This paper may be of interest in the context of a labora-tory modeling of non-stationary processes of wave-particleinteractions in space plasma, since there are a lot of openquestions about the origin of some types of emissionsin space cyclotron masers, especially mechanisms of finespectral structure [32]. Although the concept of electronholes in a phase space was occasionally used, e.g. for inter-pretation of emissions in the Earth’s auroral upward anddownward current regions [47,48], Jovian S-burst emission[49], the self-consistent theory of fast frequency events sim-ilar to those of toroidal Alfven eigenmodes in tokamaks isstill absent. We hope that the reported dynamic spectrawith frequency sweeping may eventually push the appli-cation of “holes and clumps” paradigm into a new realm.

∗ ∗ ∗

The work has been supported by RFBR (grants No.15–32–20770, 16–32–60056). The authors are grateful toProfessor Andrei Demekhov for fruitful discussions andKeysight Technologies Inc. for its technical support.

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Observation of quasi-periodic frequency sweeping in electron cyclotron emission

REFERENCES

[1] Fisch N. J. and Rax J. M., Phys. Rev. Lett., 69 (1992)612.

[2] Fisch N. J., Phys. Rev. Lett., 97 (2006) 225001.[3] Drummond W. E. and Pines D., Nucl. Fusion Suppl., 3

(1962) 1049.[4] Vedenov A. A., Velikhov E. P. and Sagdeev R. Z.,

Nucl. Fusion Suppl., 2 (1962) 465.[5] Suvorov E. V. and Tokman M. D., Plasma Phys., 25

(1983) 723.[6] Stallard B. W., Matsuda Y. and Nevins W. M.,

Nucl. Fusion, 23 (1983) 213.[7] Giruzzi G., Phys. Fluids, 31 (1988) 3305.[8] Trakhtengerts V. Y. and Rycroft M. J., Whistler

and alfven mode cyclotron masers in space (CambridgeUniversity Press, New York) 2008.

[9] Trakhtengerts V. Yu., Tagirov V. R. andChernous S. A., Geomagn. Aeron., 26 (1986) 99.

[10] Demekhov A. G. and Trakhtengerts V. Yu., J. Geo-

phys. Res., 99 (1994) 5831.[11] Trakhtengerts V. Y., Rycroft M. J. andDemekhov

A. G., J. Geophys. Res., 101 (1996) 13293.[12] Trakhtengerts V.Yu., J. Geophys. Res., 100 (1995)

17205.[13] Trakhtengerts V.Yu., Ann. Geophys., 17 (1999) 95.[14] Demekhov A.G. and Trakhtengerts V.Yu., Radio-

phys. Quantum El., 44 (2001) 103.[15] Berk H. L., Breizman B. N. and Pekker M., Phys.

Rev. Lett., 76 (1996) 1256.[16] Berk H. L., Breizman B. N. and Petviashvili N. V.,

Phys. Lett. A, 234 (1997) 213.[17] Breizman B. N., Berk H. L., Pekker M. S., Por-

celli F., Stupakov G. V. and Wong K. L., Physics of

Plasmas, 4 (1997) 1559.[18] Bernstein I. B., Greene J. and Kruskal M., Phys.

Rev. , 108 (1957) 546.[19] Lilley M. K., Breizman B. N. and Sharapov S. E.,

Phys. Plasmas, 17 (2010) 092305.[20] Nyqvist R. M., Lilley M. K. and Breizman B. N.,

Nucl. Fusion, 52 (2012) 094020.[21] Nyqvist R. M. and Breizman B. N., Phys. Plasmas, 20

(2013) 042106.[22] Lilley M. K. and Nyqvist R. M., Phys. Rev. Lett., 112

(2014) 155002.[23] Pinches S. D., Berk H. L., Gryaznevich M. P. et al.,

Plasma Phys. Control. Fusion, 46 (2004) S47.[24] Lilley M. K., Breizman B. N. and Sharapov S.

E.,Phys. Rev. Lett., 102 (2009) 195003.[25] Lesur M., Idomura Y., Shinohara K., Garbet X.,

and the JT-60 Team, Phys.Plasmas, 17 (2010) 122311.[26] Kosuga Y. and Diamond P. H., Phys. Plasmas, 19

(2012) 072307.[27] Lesur M., Phys. Plasmas, 20 (2013) 055905.[28] Hole M. J. and Fitzgerald M., Plasma Phys. Control.

Fusion, 56 (2014) 053001.[29] Lesur M., Diamond P. H. and Kosuga Y., Plasma

Phys. Control. Fusion, 56 (2014) 075005.[30] Bingham R., Speirs D.C., Kellett B.J., Vorgul I.,

McConville S.L., Cairns R.A., Cross A.W., Phelps

A.D.R. and Ronald K., Space Sci. Rev., 178 (2013)695.

[31] Van Compernolle B., An X., Bortnik J., Thorne

R.M., Pribyl P. and Gekelman W., Phys. Rev. Lett.,114 (2015) 245002.

[32] Treumann R. A., Astron. Astrophys. Rev., 13 (2006)229.

[33] Menietti J. D. et al., J. Geophys. Res.: Space Phys.,117 (2012) A12214.

[34] Trakhtengerts V. Yu., Radiophys. Quantum El., 39(1996) 463.

[35] Trakhtengerts V. Yu. and Shalashov A. G., Astron-

omy Reports, 43 (1999) 540.[36] Viktorov M. E., Golubev S. V., Zaitsev V. V. and

Mansfeld D. A., Radiophys. Quantum El., 57 (2015)849.

[37] Trigilio C. et al., Astrophys. J. Lett., 739 (2011) L10.[38] Vodopyanov A. V., Golubev S. V., Demekhov A.

G., Zorin V. G., Mansfeld D. A., Razin S. V. andShalashov A. G., JETP, 104 (2007) 296.

[39] Vodopyanov A. V., Golubev S. V., Demekhov A.

G., Zorin V. G., Mansfeld D. A., Razin S. V. andTrakhtengerts V. Yu., Plasma Phys. Rep., 31 (2005)927.

[40] Shalashov A. G., Vodopyanov A. V., Golubev S.

V., Demekhov A. G., Zorin V. G., Mansfeld D. A.

and Razin S. V. , JETP Lett., 84 (2006) 314.[41] Shalashov A. G., Golubev S. V., Gospodchikov E.

D., Mansfeld D. A. and Viktorov M. E., Plasma

Phys. Contr. F., 54 (2012) 085023.[42] Viktorov M. E., Vodopyanov A. V., Golubev S. V.,

Izotov I. V. and Mansfeld D. A., Radiophys. Quantum

El., 56 (2013) 12.[43] Viktorov M. E., Golubev S. V., Gospodchikov E.

D., Izotov I. V., Mansfeld D. A. and Shalashov A.

G., Radiophys. Quantum El., 56 (2013) 216.[44] Viktorov M. E., Mansfeld D. A. and Golubev S.

V., Eur. Phys. Lett., 109 (2015) 65002.[45] Vorgul I., Cairns R. A. and Bingham R., Phys. Plas-

mas, 12 (2005) 122903.[46] Trakhtengerts V. Y., The relaxation of plasma with

anisotropic velocity distribution, in Basic Plasma Physics,edited by Galeev A. A., Sudan R. N., Vol. 2 (North-Holland Physics Publishing, Amsterdam) 1984, p. 519.

[47] Treumann R. A., Baumjohann W. and Pottelette

R., Ann. Geophys., 29 (2011) 1885.[48] Treumann R. A., Baumjohann W. and Pottelette

R., Ann. Geophys., 30 (2012) 119.[49] Goodrich K. A. and Ergun R. E., Astrophys. J., 809

(2015) 4.

p-7