Phys. Plasmas 26, 032105 (2019); https://doi.org/10.1063/1.5082802 26, 032105

© 2019 Author(s).

Observation of spontaneous decay ofAlfvénic fluctuations into co- and counter-propagating magnetosonic waves in alaboratory plasmaCite as: Phys. Plasmas 26, 032105 (2019); https://doi.org/10.1063/1.5082802Submitted: 24 November 2018 . Accepted: 22 February 2019 . Published Online: 14 March 2019

Peiyun Shi , Zhida Yang , Ming Luo, Rongsheng Wang, Quanming Lu, and Xuan Sun

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Observation of spontaneous decay of Alfv�enicfluctuations into co- and counter-propagatingmagnetosonic waves in a laboratory plasma

Cite as: Phys. Plasmas 26, 032105 (2019); doi: 10.1063/1.5082802Submitted: 24 November 2018 . Accepted: 22 February 2019 .Published Online: 14 March 2019

Peiyun Shi,1 Zhida Yang,1 Ming Luo,1 Rongsheng Wang,2 Quanming Lu,2 and Xuan Sun1,a)

AFFILIATIONS1Department of Engineering and Applied Physics, University of Science and Technology of China, Hefei 230026, China2Department of Geophysics and Planetary Science, University of Science and Technology of China, Hefei 230026, China

a)Email: xsun@ustc.edu.cn

ABSTRACT

We show experimentally that Alfv�enic fluctuations can spontaneously decay into a fast and a slow magnetosonic wave in an inhomogeneousplasma. The fast wave of higher frequency propagates in the same direction, while the slow wave of lower frequency propagates in theopposite direction with the pump wave. Both of the daughter waves are characterized by strong parallel but modest perpendicularfluctuations. The measured frequencies and wavenumbers are found to satisfy the energy and momentum conservation conditions for anonlinear three wave interaction. The evidence of energy flowing from pump fluctuations into daughter waves is also presented in this paper.The results may shed light on the origin of inward Alfv�en waves observed in the solar corona and chromosphere and how shear Alfv�enwaves deposit its energy by driving compressional perturbations.

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I. INTRODUCTION

The laboratory simulation of space plasma has continuouslyenriched our understanding of plasma behavior in the planetary andstellar atmosphere, for example, magnetic reconnection, wave-particleinteraction, and magneto-rotational instability.1–4 As the most funda-mental wave mode in magnetized plasmas, Alfv�en waves could alsoplay a role in the heating and evolution of space, astrophysical, andlaboratory plasmas.5 Many experiments have been thus devoted to thestudy of Alfv�en waves and revealed many faces of Alfv�en waves.6–9

Due to their incompressible nature, shear Alfv�en waves (SAWs)have been considered as the primary carrier of energy to the coronalregion; on the other hand, there must exist a mechanism(s) to converttheir energy into heat which accounts for the mysterious high temper-ature of the corona. The conversion does not need to be very efficientas a small amount (�0.1%) of output energy from the interior isenough to heat up the tenuous corona.10 Of various hypotheses, twocandidates are of particular interest here: (1) parametric excitation ofion sound waves11,12 or other compressional waves13 and (2) turbulentheating from a nonlinear cascade of multiple wave interactions.14–16

Although related theoretical and numerical investigations17,18 of spaceplasma have been conducted extensively, few laboratory observationsof simulating plasmas have been reported. By deliberately launching

two Alfv�en waves at different frequencies, ion sound waves with differ-ential frequency were successfully launched through beating of thesetwo waves in Large Plasma Devices (LAPDs).19,20 Recently, Dorfmanet al.21 observed that a kinetic Alfv�en wave can produce two sidebandsof Alfv�en waves and a low frequency non-resonant mode due to mod-ulational instability. We will show in this paper that shear Alfv�enicfluctuations (SAFs) could also spontaneously decay into co- andcounter-propagating magnetosonic waves in a laboratory plasma, andthe decay channel is robust. Note that this process has been consideredin theories;13,22–25 however, this kind of decay channel has not beenobserved in a laboratory plasma, to the authors’ knowledge.

The turbulent cascade of non-compressional waves requires acounter-propagating wave.15,26 Although the magnetic structures in solarplasmas have been frequently inspected by satellite imaging and spectro-scopic systems, the identification of the wave’s direction has posed chal-lenges. Recently, Mortan et al.27 reported the observation of inwardAlfv�en waves in the coronal region and Liu et al.28 found the counter-propagating compressive fluctuation in the chromosphere. Notably, aninward propagating wave can be generated by the wave reflection orparametric decay, while in this paper, we show that counter-propagating(inward) magnetic waves can be produced by the latter process.

The rest of this paper is arranged as follows: Sec. II introduces theexperimental setup including the SAF launching antenna and

Phys. Plasmas 26, 032105 (2019); doi: 10.1063/1.5082802 26, 032105-1

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diagnostic tools. The energy and momentum conservation constrainsare shown to be satisfied in Sec. III to validate the three wave interac-tion during this spontaneous decay process of SAFs. The identificationof daughter wave modes is also provided in this section. Section IVshows the evidence of the energy transferring process. Finally, Sec. Vpresents a brief discussion and the conclusion of this paper.

II. EXPERIMENTAL SETUP

The experiments were conducted using a 10-m long tandemmirrordevice, KMAX (Keda Mirror with AXisymmetricity), which consists ofone central cell and two end cells.29 Their dimensions are shown in Fig.1(a), and the green dashed lines represent the DC magnetic field, point-ing toward þz, which is typically 275 gauss in the central cell and 1900gauss at the mirror throat. In this work, the plasma is produced by a heli-con antenna located at the right end, z � 5:05m, the RF power is typi-cally 2 kW with frequency at 13.56MHz, and the gas species ishydrogen. Helicon plasmas have been widely used in space plasma simu-lation due to their easy operation and high ionization rate.30 In ourapplication, the source plasma expands into downstream by followingthe background magnetic field lines. The key to the experimentsreported here is to form a plasma column through the entire device, i.e.,

a 10 m long plasma column. Typical plasma parameters arene � ð1� 4Þ � 1016 m�3 andTe � 4� 10 eV: Radial density profilesmeasured by four triple Langmuir probes PE1–4 at four different axialpositions are given in Fig. 1(b). The SAF is launched by a solenoidalantenna at z ¼ 1:00m and r¼ 0.05 m. A nominal 7 kW oscillator at afrequency of f¼ 368kHz or x� 0.89 Xci, where Xci is the ion cyclotronfrequency, feeds current into the antenna via an isolation transformer.The antenna current measured using a Pearson Current Monitor is�100A, which can produce the magnetic field Bx of�20 gauss, 7.3% ofthe background magnetic field strength. Four sets of electrically shieldedmagnetic probes PBE1, PBE2, PBW1, and PBW2 are placed at z¼ 0.00m, 0.50 m, 1.67 m, and 3.25 m, respectively, to detect magnetic fluctua-tion. PBW1 also integrates a single electric probe at the top to measurethe ion saturation currents. All signals from probes are connected tooscilloscopes with a sampling rate at 5 MS/s.

III. OVERVIEW OF THE SPONTANEOUS DECAYPHENOMENONA. Energy conservation

For any nonlinear three wave interaction, energy and momentumhave to be conserved: f¼ f1 þ f2 and k ¼ k1 þ k2. These conservation

FIG. 1. (a) Machine diagram of the KMAXtandem mirror device. Hydrogen plasma isgenerated by a Nagoya type III antennalocated at z¼ 5.05 m. The green dashedlines indicate the DC magnetic field linestowards þz; the brown triangles indicatethe positions of triple probes PE1-4 fromright to left; the blue circles indicate themagnetic probes PBE1, PBE2, PBW1,and PBW2 at z¼ 0.00 m, 0.50 m, 1.67 m,and 3.25 m; and the red diamond indi-cates the Alfv�en launching antenna atr¼ 0.05 m. The black arrows at the bot-tom illustrate the axial current of the waveinside plasma at one moment, and thegrey arrows indicate the return currentpossibly through endplates and vesselwall. (b) Radial density profiles at z¼ 4.05m, 1.75 m, �0.50 m, and �3.10 m, mea-sured by PE1-4, respectively.

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constrains are presented in Figs. 2 and 3, respectively. Clearly, thepump fluctuation at f ¼ 368 kHz has driven two daughter waves atf1 ¼ 311 kHz and f2 ¼ 57 kHz. Figure 2(a) displays the spectrum ofoutput current from the oscillator whose Q is 35 000, high enough tobe considered as a monochromatic emitter. Figures 2(b)–2(d) showthe measured magnetic fluctuation spectra in the x, y, and z directions,respectively, and Fig. 2(e) shows the spectrum of ion saturation cur-rent collected by a single probe, i.e., one tungsten tip, integrated withinmagnetic probe suite PBW1. The launched fluctuation has beendetected largely in the x direction, with an amplitude of �0.12 gaussor�0.04% of background field strength, which is in the same directionwith the oscillating magnetic fields generated by this solenoidalantenna. Rather than launching a wave with a defined wavenumberand direction, this antenna more likely just perturbs the plasma orenhances the thermal fluctuations which are in phase with the drivingoscillations. Hence, the wave may not be a single mode. In the low fre-quency limit, the degeneracy of torsional Alfv�en waves and compres-sional waves can produce linearly polarized waves. It may explain whythe dBx component dominates in our measurements. The perturbationshould travel in both directions; however, only the þz directionalpropagation is found in our phase shift measurements [see Fig. 3(a)for the data]. Considering that the endplates of our device are conduc-tors and the length of the wave is comparable with the device, the axialcurrent of the wave can exit the plasma from one plate and re-enterfrom the other end in order to close its current loop in both paralleland perpendicular directions. In other words, the periodical boundarycondition in our experiment allows the traveling wave to exist.

The most prominent feature of Fig. 2 is the presence of f1 and f2on the z directional magnetic probe and electric probe. Those twowaves only appear when both of the source oscillator and the heliconplasma are turned on. Different from the SAF discussed above, themagnetosonic wave must be accompanied by dBz and density

fluctuations for its compression nature, as observed in our experiment[see Figs. 2(d) and 2(e)]. The dBz components are 1.0–2.0 gauss instrength and�0.3%–0.7% of the background field.31

B. Momentum conservation

In addition to energy conservation, the momentum must be con-served too, i.e., k ¼ k1 þ k2, which can be derived from the phase differ-ence between magnetic probes. Figures 3(a) and 3(b) show their phasemeasurements in parallel and perpendicular directions, respectively, andthe calculated kk1 ¼ 1:1560:15 rad/m and kk2 ¼ �0:3060:10 rad/m. The non-zero kk1;2 of these compressive magnetic modes (dBz) dic-tates the existence of non-zero perturbations in the perpendicular direc-tion. This is consistent with the observation of the very modest dBx and/or dBy fluctuations at f1 and f2 in Figs. 2(b) and 2(c).

Figure 3(c) shows the parallelogram in the (x; kk) plane reflect-ing the resonant conditions for parametric decay. Here, we denote the“þ” sign to the wave propagating along the background field linedirection, i.e.,þz. This fluctuation propagating in theþz direction hasa phase velocity ð4:160:4Þ � 106 m/s, within the error of the pre-dicted value ð5:061:3Þ � 106 m/s from the SAW dispersion consider-

ing the Hall MHD effect, x ¼ kkVA

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� x2=Xci

2p

, where VA is theAlfv�en speed.

Clearly, the mother fluctuation (f ¼ 368 kHz, kk ¼ 0:5560:05rad/m, and k? ¼ 062 rad/m) has decayed into a forward propagatingwave (f1 ¼ 311kHz, kk1 ¼ 1:1560:15 rad/m, and k?1 ¼ �1264rad/m) and a backward propagating wave (f2 ¼ 57kHz, kk2 ¼ �0:3060:10 rad/m, and k?2 ¼ 1062 rad/m), satisfying both energy andmomentum conservation equations. Note that there have been extensivetheoretical and simulation works on the parametric decay of SAWs intoAlfv�en and ion sound waves to explain the coronal heating, and ourresults have verified that there exists a channel for Alfv�en fluctuations todecay into fast and slow waves. Moreover, it also shows that the para-metric decay can produce the backward magnetic wave, a key ingredientfor the turbulent cascade. Although not shown here, the measuredamplitudes of waves are found to decrease with the radius, only�0.1 Gnear the edge.

C. Identification of wave modes

1. Dispersion relation

A two-fluid description of low frequency magnetic waves is givenby Ref. 5

x2ð1þ k2de2Þ � VA

2kk2

h i� x2ð1þ k2de

2Þðx2 � Cs2k2Þ

�

�VA2k2ðx2 � Cs

2kk2Þ�¼ x2VA

4k2kk2ðx2 � Cs

2k2Þ=Xci2; (1)

where CS is the sound speed and de is the electron inertial length. Thefirst term describes the Alfv�en and the second term magnetosonicwaves. By substituting the experimental values of CS ¼ 3:0� 104 m/s, VA ¼ 2:5� 106 m/s, and de ¼ 0:04 m in Eq. (1), we plot the rela-tionship between x and kk for different jk?j in Fig. 3(d) as color-coded lines, along with the measured data of f1 and f2 waves assquares. Comparing with the model prediction, it is found the f1 wavefits much better than the f2 wave because k? is very sensitive to kk inthe low frequency regime.

FIG. 2. Frequency spectrum of the Alfv�en antenna current, magnetic field, and den-sity fluctuation: (a) the Alfv�en antenna emits waves at f¼368 kHz; (b) the dBxspectrum shows a peak at 368 kHz and also two small but distinguishable peaks atf1 ¼ 311 kHz and f2 ¼ 57 kHz, which can be clearly seen on the two inset panels;(c) the dBy spectrum shows a small peak at f1 but a negligible or questionablepeak at f2 and no discernible peak at 368 kHz; (d) the dBz spectrum shows clearlythe spontaneously decayed daughter waves resonating at 57 and 311 kHz and nopeak at 368 kHz; and (e) the spectrum of plasma density fluctuation shows twopeaks at 57 and 311 kHz.

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2. Correlation between dne and dBz

To further identify specific modes of these magnetosonic waves,we adopt the common approach used in space plasma. The correlationbetween density and parallel magnetic fluctuation is positive for thefast mode and negative for the slow mode.32 In other words, dne anddBz are in phase for fast magnetosonic waves and out of phase forslow magnetosonic waves.5

We can obtain the phase difference between dne and dBz from thenormalized Cross-Spectral Density (CSD) of measured dIsat and daugh-ter waves’ dBz , a¼ CSDðdIsat; dBzÞ=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPSDðdIsatÞ � PSDðdBzÞ

p, where

PSD stands for the Power Spectral Density. The result is shown in Fig. 4,where the red dotted line represents the degree of coherence aj j2 and theblue solid line represents the phase difference, arctan ðaÞ. To have amore accurate estimation, we average the phase difference over a smallfrequency range whose coherence value is larger than 0.85, as shown inthe grey region in Fig. 4, and the results are 35�64� for the f1 wave and150�66� for the f2 wave. Adding the additional phase shift due to theradial separation of magnetic and electric probes,�ð16�68�Þ for f1 andþð12�66�Þ for f2, the phase differences are found to be 19�614� for f1(311kHz) and 162�628� for f2 (57kHz), respectively. The errors esti-mated above have included the errors due to the electron temperaturefluctuation. Thus, it is reasonable to conclude that f1 (in phase) is thefast and f2 (out of phase) is the slowmagnetosonic wave.

IV. ENERGY TRANSFERRING PROCESS

The energy transferring process is studied by analyzing the waveamplitudes as a function of time. Figure 5 shows that the SAF starts togrow at t� 50 ls, while two daughter waves are not present until

t� 200 ls. This time delay reflects that a minimum strength of thepump wave is required to generate the other two waves.33–35 A directevidence of the energy transferring from the SAF to two daughterwaves is shown as the dashed green line in Fig. 5(b), the SAF ampli-tude normalized to antenna current, which shows a relatively flatregime between t¼ 50 ls and t¼ 200 ls and starts to drop when twodaughter waves appear.

FIG. 3. (a) The phase shift relative to theantenna or PBE1, measured by four mag-netic probes at different z’s, with the redsquare, black circle, and blue triangle forwaves f , f1; and f2, respectively. (b) Theperpendicular phase shift versus x, andthe symbols are used in the same manneras (a). (c) Parallelogram in the (x; kk)plane reflecting the resonant conditionsfor parametric decay. The shaded errorbars indicate the uncertainty in kk. (d)Theoretical prediction for different jk?j,and measured data, f1 and f2, are indi-cated as squares.

FIG. 4. Phase difference between dne and dBz fluctuation. The cross-spectral den-sity is plotted as blue solid line, while the corresponding coherence is shown as thered dotted line. The grey shades indicate the credible regions, where the coherencevalue must be greater than 0.85, for averaging the phase difference: 35�64� forwave f1 and 150

�66� for wave f2.

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V. CONCLUSION

A possible explanation for parametric decay to occur in thisexperiment is through the ponderomotive force.35 The SAF and mag-netosonic wave, or just thermal fluctuations in the initial stage, cancouple together through the nonlinear force djy � dBz

� �, where djy and

dBz are the polarization current of the SAF and parallel magnetic fieldcomponent of one daughter wave. Then, the force could influence orenhance the plasma velocity component dvx of differential frequencycorresponding to another daughter wave, which leads to density com-pression and rarefaction via term k?� dvx . If the dispersion relations ofthese waves are satisfied by the experimental plasma parameters, twodaughter waves could be fed continuously by pump waves and growto an appreciable level.

In summary, we have presented experimental demonstration thata shear Alfv�en fluctuation can spontaneously decay into a co-propagating fast magnetosonic wave and a counter-propagating slowmagnetosonic wave. The constraints of energy and momentum con-servation are satisfied. The modes of daughter waves are validated bycomparison with a dispersion relation and by correlation between den-sity and magnetic field fluctuation. Both of the daughter waves can beeffectively damped, leading to the dissipation of wave energy. The slowmagnetosonic wave has been widely deemed as a candidate for theheating of the coronal region. Similar to the observation of inwardpropagating waves in the coronal and chromosphere region, we alsoobserved the counter-propagating wave in the experiment. Whether itmay lead to the turbulent heating is a subject to the future studybecause the current source power is not high enough to see this effect.In a word, the experiment reported in this paper has opened a newwindow to study space plasma relevant physics, and it may also help

to utilize Alfv�en wave as an efficient supplementary heating methodvia the parametric decay process.

ACKNOWLEDGMENTS

This work was supported by the National Key R&D Program ofChina under Contract Nos. 2017YFE0301802 and 2017YFA0402500,the National Natural Science Foundation of China under Grant No.11475172, and the Key Research Program of Frontier Sciences, CASunder Grant No. QYZDJ-SSW-DQC010.

REFERENCES1M. E. Koepke, Rev. Geophys. 46, RG3001, https://doi.org/10.1029/2005RG000168(2008).

2E. M. Tejero, C. Crabtree, D. D. Blackwell, W. E. Amatucci, M. Mithaiwala, G.Ganguli, and L. Rudakov, Phys. Plasmas 22, 091503 (2015).

3M. Yamada, R. Kulsrud, and H. Ji, Rev. Mod. Phys. 82, 603 (2010).4H. Ji, M. Burin, E. Schartman, and J. Goodman, Nature 444, 343 (2006).5N. F. Cramer, The Physics of Alfv�en Waves (Wiley, 2011).6W. Gekelman, S. Vincena, B. Van Compernolle, G. J. Morales, J. E. Maggs, P.Pribyl, and T. A. Carter, Phys. Plasmas 18, 055501 (2011).

7S. Houshmandyar and E. E. Scime, Phys. Plasmas 18, 112111 (2011).8D. Leneman, W. Gekelman, and J. Maggs, Phys. Rev. Lett. 82, 2673 (1999).9C. Watts and J. Hanna, Phys. Plasmas 11, 1358 (2004).

10R. Erdelyi and I. Ballai, Astron. Nachr. 328, 726 (2007).11R. Z. Sagdeev and A. A. Galeev, Lectures on the Non-Linear Theory of Plasma(W.A. Benjamin, New York, 1969).

12L. Del Zanna, M. Velli, and P. Londrillo, Astron. Astrophys. 367, 705 (2001).13S. S. Moiseev, V. N. Oraevsky, and V. G. Pungin, Non-Linear Instabilities inPlasmas and Hydrodynamics (Taylor & Francis, 1999).

14R. H. Kraichnan, Phys. Fluids 8, 1385 (1965).15P. Goldreich and S. Sridhar, Astrophys. J. 438, 763 (1995).16S. Sridhar and P. Goldreich, Astrophys. J. 432, 612 (1994).17X. Gao, Q. Lu, X. Li, Y. Hao, X. Tao, and S. Wang, Astrophys. J. 780, 56 (2013).18X. Gao, Q. Lu, X. Li, L. Shan, and S. Wang, Phys. Plasmas 20, 072902 (2013).19G. G. Howes, D. J. Drake, K. D. Nielson, T. A. Carter, C. A. Kletzing, and F.Skiff, Phys. Rev. Lett. 109, 255001 (2012).

20S. Dorfman and T. A. Carter, Phys. Rev. Lett. 110, 195001 (2013).21S. Dorfman and T. A. Carter, Phys. Rev. Lett. 116, 195002 (2016).22V. N. Oraevsky, Nucl. Fusion 4, 263 (1964).23K. S. Karplyuk and V. N. Oraevskii, JETP Lett. 5, 365 (1967).24F. F. Cap, Handbook on Plasma Instabilities (Academic Press, 1982).25G. Brodin and L. Stenflo, J. Plasma Phys. 39, 277 (1988).26S. Boldyrev, Phys. Rev. Lett. 96, 115002 (2006).27R. J. Morton, S. Tomczyk, and R. Pinto, Nat. Commun. 6, 7813 (2015).28Z. X. Liu, J. S. He, and L. M. Yan, Res. Astron. Astrophys. 14, 299 (2014).29Q. Zhang, P. Shi, M. Liu, M. Lin, and X. Sun, Fusion Sci. Technol. 68, 50(2015).

30J. L. Kline, E. E. Scime, P. A. Keiter, M. M. Balkey, and R. F. Boivin, Phys.Plasmas 6, 4767 (1999).

31L. Matteini, S. Landi, L. Del Zanna, M. Velli, and P. Hellinger, Geophys. Res.Lett. 37, L20101, https://doi.org/10.1029/2010GL044806 (2010).

32G. G. Howes, S. D. Bale, K. G. Klein, C. H. K. Chen, C. S. Salem, and J. M.TenBarge, Astrophys. J. Lett. 753, L19 (2012).

33C. S. Liu and V. K. Tripathi, Phys. Rep. 130, 143 (1986).34J. L. Kline and E. E. Scime, Phys. Plasmas 10, 135 (2003).35F. F. Chen, Introduction to Plasma Physics and Controlled Fusion (Springer,2006).

FIG. 5. Time history of (a) antenna current, (b) SAF and normalized SAF, (c) fastwave, and (d) slow wave amplitude from t¼ 0 to 1000 ls. The SAF amplitude nor-malized by antenna current, plotted as a green dashed line in panel (b), decreasesat t¼ 200 ls, which indicates that SAF energy is being injected into the daughtermodes.

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