Astrophys Space Sci (2017) 362:172 DOI 10.1007/s10509-017-3156-2

O R I G I NA L A RT I C L E

Nonlinear whistler wave model for lion roars in the Earth’smagnetosheath

N.K. Dwivedi1 · S. Singh2

Received: 14 February 2017 / Accepted: 7 August 2017© Springer Science+Business Media B.V. 2017

Abstract In the present study, we construct a nonlinearwhistler wave model to explain the magnetic field spectraobserved for lion roars in the Earth’s magnetosheath region.We use two-fluid theory and semi-analytical approach to de-rive the dynamical equation of whistler wave propagatingalong the ambient magnetic field. We examine the magneticfield localization of parallel propagating whistler wave inthe intermediate beta plasma applicable to the Earth’s mag-netosheath. In addition, we investigate spectral features ofthe magnetic field fluctuations and the spectral slope value.The magnetic field spectrum obtained by semi-analytical ap-proach shows a spectral break point and becomes steeper athigher wave numbers. The observations of IMP 6 plasmawaves and magnetometer experiment reveal the existenceof short period magnetic field fluctuations in the magne-tosheath. The observation shows the broadband spectrumwith a spectral slope of −4.5 superimposed with a narrowband peak. The broadband fluctuations appear due to the en-ergy cascades attributed by low-frequency magnetohydro-dynamic modes, whereas, a narrow band peak is observeddue to the short period lion roars bursts. The energy spec-trum predicted by the present theoretical model shows a sim-ilar broadband spectrum in the wave number domain with aspectral slope of −3.2, however, it does not show any nar-row band peak. Further, we present a comparison betweentheoretical energy spectrum and the observed spectral slopein the frequency domain. The present semi-analytical modelprovides exposure to the whistler wave turbulence in theEarth’s magnetosheath.

B S. Singhssi@et.aau.dk

1 Space Research Institute, Austrian Academy of Sciences, 8042Graz, Austria

2 Department of Energy Technology, Aalborg University,Pontoppidanstræde 111, Aalborg East 9220, Denmark

Keywords Whistler wave · Lion roars · Nonlinearphenomenon · Turbulence · Magnetosheath · Power-laws

1 Introduction

Amongst several plasma waves detected in the Earth’s mag-netosheath region, whistler waves are the dominant wavesin a high-frequency range from 100 Hz to 1000 Hz. Thestudy of magnetic field fluctuations in the Earth’s magne-tosheath has been of a great interest to understand the spaceplasma phenomena associated with the Earth’s bow shockwaves and magnetopause such as turbulence, particle accel-eration in the boundary regions. Moreover, the interpretationof these processes ambiguously differs from one observationto the other observation mainly due to a continuous evolu-tionary state.

The magnetosheath plasma waves propagating nearlyparallel to the ambient magnetic field referred as lion roars(as realized by wave packet) are first reported by Smithet al. (1967, 1969). Lion roars are identified as resonantlyamplified whistler waves driven by thermal electrons dueto temperature anisotropy (Smith et al. 1967, 1969; Smithand Tsurutani 1976). These waves are intense, narrow bandtones with magnetic field fluctuations in the frequency range100–300 Hz. Due to nonlinear effects, their generation in themagnetosheath is generally reported as a complex mecha-nism. Throne and Tsurutani (Thorne and Tsurutani 1981;Tsurutani et al. 1982) study the generation mechanism forthe magnetosheath lion roars on the basis of ISEE satel-lite observational data, and show that lion roars are gener-ated due to an electron cyclotron instability caused by thepresence of anisotropic magnetosheath electrons with per-pendicular temperature greater than the parallel temperature.Moreover, long duration whistler waves with a lasting time

172 Page 2 of 7 N.K. Dwivedi, S. Singh

of more than 5 minutes in the frequency range similar to thelion roars are observed by Rodriguez (1985).

Zhang et al. (1998) use the waveform capture instru-ment on-board Geotail (Nishida 1994) to study 20–300 Hzlion roars with a typical amplitude of 0.1 nT within themirror mode (type A). Additionally, similar waves are ob-served without ambient magnetic field depletion close to thebow shock (type B). They observe only 30% of the lionroars (type A) related to a dip in the magnetic field. Whiletype B waves do not show any correlation between the oc-currence of lion roars and the ambient magnetic field. Fur-thermore, Tsurutani et al. (1982) discuss the source of freeenergy for the whistler mode waves. The shock compres-sion plus magnetic field draping around the magnetospherelead to a greater T⊥/T‖ instability. The similar anisotropyis noted for the protons which results in the mirror insta-bility. This mirror instability generates the lion roars prop-agating primarily along the magnetic field. The high den-sity mirror mode waves guide whistlers by ducting, and thusthe waves, in general, kept propagating in the parallel direc-tion. Baumjohann et al. (1999) examine the high resolutiondata of 128 Hz from Equator-S magnetometer and investi-gate the waveform of the mirror mode through lion roars.Masood et al. (2006) for the first time compare the theoryand data for the source region of lion roars that are not as-sociated with the mirror waves. They suggest the source oflion roars can be local or remote, however, in the majorityof the cases the source is somewhere close to the bow shockwhere the field is less compressed. Particle reflection at thebow shock and the ion foreshock accounts for the upstreamsources of turbulence and free energy to initiate local insta-bilities.

It is worthwhile noting that the magnetospheric chorusis quite similar to whistler mode wave and is generatedby an electron temperature anisotropy (Palmadesso and Pa-padopoulos 1979). The waves are nearly parallel propagat-ing and sometimes have coherent wave packets (Santolíket al. 2003; Tsurutani et al. 2009, 2011) at large angles(Goldstein and Tsurutani 1984). This similarity between thechorus and lion roars has led many observers to believe thatthey are the same waves. Maksimovic et al. (2001) performa spatio-temporal analysis of magnetic field fluctuations todetermine the location of the source region, and character-istics of the magnetosheath lion roars. They discover sev-eral lion roars which propagate in opposite directions whichis consistent with the study of Zhang et al. (1998). In thelight of above discussion, understanding of whistler turbu-lence in the presence of density fluctuations is not only im-portant but also decisive in the context of nonlinear wavesand turbulence in the solar wind plasma streaming outwardsfrom the Sun (Bhattacharjee et al. 1998; Stawicki et al.2005; Krafft and Volokitin 2003; Ng et al. 2003; Vockset al. 2005; Gary et al. 2008; Saito et al. 2008; Dwivedi

et al. 2012; Dwivedi and Sharma 2013), magnetic reconnec-tion in the Earth’s magnetosphere (Wei et al. 2007; Sharmaet al. 2010), turbulence in the interstellar medium, which isstirred by violent events like supernova explosions (Burman1975), and astrophysical plasmas (Roth 2007) where charac-teristic fluctuations can naturally be of several astronomicalunits. A kinetic description is typically needed for high-βanisotropic conditions. Fluid theories provide crucial insightinto the wave properties, magnetic field turbulence spectrumand corresponding spectral slopes, however, cannot describeplasma instabilities which arise at the kinetic scale and im-part misleading wave characteristics.

Space turbulence is recognized as one of the most funda-mental problem of plasma physics that has not yet been fullyunderstood. Turbulence commonly occurs in the physicalsystem of nonlinear dispersive waves. The energy transferbetween waves occurs mostly among resonant sets of waves.In most of the cases, the nonlinearity is small and, there-fore, dispersive wave interactions are weak. These mutualinteractions among the waves are frequently described bythe weak turbulence theory. The weak turbulence theory hasbeen extensively developed between 1950–1970 (Kadomt-sev 1965; Sagdeev and Galeev 1969; Vedenov 1968; David-son 1972; Hasegawa 1975; Akhiezer et al. 1975) to describethe transfer of energy between different frequencies withweak nonlinearities (Yoon 2000; Yoon et al. 2012, 2016).Irrespective of the extensive application, the weak turbu-lence theory ignores particle response to the presence andmutual interaction of the waves. The strong turbulence the-ory, however, is qualitative since the basic equation cannotbe analytically solved unless reduced to the Korteweg-deVries or the nonlinear Schrödinger equation when closedsolutions exist for the stationary state. These solutions de-scribe solitary wave structures. These solitary structures traphigh-frequency waves, but are themselves low frequencywaves.

In the past, most of the studies focused on the observa-tional aspects of the lion roars e.g. waveform, angle of prop-agation (Cornilleau-Wehrlin et al. 1997; Baumjohann et al.1999; Smith and Tsurutani 1976), frequency range (Smithand Tsurutani 1976), and turbulence properties associatedwith the lion roars (Rodriguez 1985). So far, very few stud-ies, discussing the turbulence processes, spectral featuresof the magnetic fluctuations associated with lion roars andtheir comparison with the observational results in the light ofknown theoretical models are available. The work reportedby Masood et al. (2006) and Qureshi et al. (2014) is mainlyfocused on the source region of the lion roars and on the gen-eration mechanism. However, the prime intent of the presentpaper is to envisage the nonlinear turbulence mediated bywhistler waves in the presence of density perturbations andto realize how the energy is being transported from large tosmall scale in the Earth’s magnetosheath. Here, we propose

Nonlinear whistler wave model for lion roars in the Earth’s magnetosheath Page 3 of 7 172

a two-fluid model of a nonlinear whistler wave propagationto examine the magnetic field localized structures and en-ergy spectrum in the presence of the ponderomotive force.The ponderomotive nonlinearity modifies the backgrounddensity of plasma and creates a density channel. A semi-analytical method is developed to solve the model equationand exemplify the wave localized structure due to pondero-motive nonlinearity. We also estimate the energy spectrumof the wave magnetic field and compare spectral index withthe observational results (Rodriguez 1985).

The paper is organized in the following sections: Sec-tion 2 presents intermediate steps of the derivation part ofthe dynamical equation of whistler wave, and a brief de-scription of the adopted semi-analytical approach. Section 3presents the results obtained by the semi-analytical approachand physical insights on the nonlinear processes leading tothe turbulence in the magnetosheath region. Section 4 con-cludes with the results, limitations and future recommenda-tions of the present work.

2 Two-fluid model and wave dynamics

In the present model, we consider homogeneous collision-less plasma consisting of ions and electrons. In order to an-alyze the nonlinear evolution of the whistlers, we adopt thetwo-fluid model of plasma for the steady state. We derive thedynamical equations that govern whistler wave propagationalong the background magnetic field. The model equationsare numerically solved to realize the turbulence spectrum.

2.1 Whistler wave coupled dynamics

We start with the wave equation for the electric field wherethe ambient magnetic field is along the z-axis, �B0 = B0z andthe electric vector, �E = E exp(iωt),

∇2 �E − �∇( �∇ · �E) + ω2

c2ε · �E = 0 (1)

where ε is the dielectric tensor, and ω is the whistler fre-quency.

The wave equations can be written in component formas,

∂2Ex

∂z2− ∂2Ez

∂x∂z= −ω2

c2(ε · �E)x (2)

∂2Ey

∂z2+ ∂2Ey

∂x2= −ω2

c2(ε · �E)y (3)

We solve the wave equations with an assumption that theelectric field variation along the background magnetic field(z-axis) is stronger than the variation in the xy-plane. The

wave is thus, treated as transverse in the zeroth order ap-proximation and no space charge is generated in the plasma( �∇ · �D) = 0.

∂Ez

∂z= 1

εzz

[εxx

∂Ex

∂x+ εxy

∂Ey

∂x

](4)

We define two coupled modes as,

EL = Ex + iEy, ER = Ex − iEy (5)

where EL and ER denote the left and right circularly po-larized modes. Assuming Ez smaller than Ex and Ey , and∂∂y

= 0, we eliminate Ez in terms of Ex and Ey by usingthe z component of the wave equation. From Eqs. (2), (3)and (5), we get the equations for the left and right circularlypolarized modes as,

∂2EL

∂z2+ 1

2

(1 + ε+00

ε00

)∂2EL

∂x2+ 1

2

(1 + ε−00

ε00

)∂2ER

∂x2

+ ω2

c2ε+EL = 0 (6)

∂2ER

∂z2+ 1

2

(1 + ε−00

ε00

)∂2ER

∂x2+ 1

2

(1 + ε+00

ε00

)∂2EL

∂x2

+ ω2

c2ε−ER = 0 (7)

where ε0 = 1 − ω2pe(1+ δns

n0)

ω2 , ε+ = 1 − ω2pe(1+ δns

n0)

ω(ω+ωce), ε− = 1 −

ω2pe(1+ δns

n0)

ω(ω−ωce), ε+00, ε−00 are the linear part of the dielectric

tensor. δns = ne − n0, where ne is the perturbed numberdensity, n0 is the unperturbed background number density,δns

n0= exp(α0| ER |2) − 1, where α0 = ω2

pi

32πn0T (ω2ci−ω2)

(1 −ω

ωci), ωce = eB0

mecis the electron cyclotron frequency, ωpe =√

4πn0e2

meis the electron plasma frequency, and me is the

mass of electron.As the main objective of the present work is to investigate

the magnetic field turbulence spectrum of the whistler wave,which is right circularly polarized wave, we consider onlyright circularly polarized wave mode i.e. �ER in Eq. (7) as,

∂2ER

∂z2+ 1

2

(1 + ε−00

ε00

)∂2ER

∂x2+ ω2

c2ε−ER = 0 (8)

2.2 Semi-analytical approach

We consider a generalized plane wave solution for Eq. (8)as,

�ER = E exp i(k‖z − ωt) (9)

172 Page 4 of 7 N.K. Dwivedi, S. Singh

where k‖ = ωcε

12−00 and E is the wave amplitude. Substitut-

ing Eq. (9) in Eq. (8), we get,

2ik‖∂E

∂z+ 1

2

(1 + ε−00

ε00

)∂2E

∂x2+ ω2

c2ε−E

− k2‖E = 0 (10)

We introduce an additional eikonal solution S(x),

E = E0(x) exp[i(S(x) + k‖z

)](11)

and split Eq. (10) into real and imaginary parts as,

2k2‖(E0)2 ∂S

∂z+ E0

2

(1 + ε−00

ε00

)[∂2E0

∂x2− k2−E0

(∂S

∂x

)2]

+ ω2

c2ε−(E0)

2 − k2−E0 = 0 (12)

2k‖E0∂E0

∂z+ E0

2

(1 + ε−00

ε00

)[2k‖∂E0

∂x

∂S

∂x− k‖E0

∂2S

∂x2

]

= 0 (13)

In order to obtain an analytical expression for the waves, weset a specific form using an assumption of Gaussian beamas,

E02 = (E0)2

f0exp

(− x2

r20 f 2

0

)(14)

S(x) = x2

(1 + ε−00ε00

)f0

df0

dz(15)

β(z) = 2

(1 + ε−00ε00

)f0

df0

dz(16)

where f0 is the dimensionless beam width parameter forwhistler wave and r0 is the transverse scale of whistler wave.Both eikonal S(x) and phase β(z) are functions of the beamwidth parameter f0 and the density variation δns . In thelimit of paraxial approximation i.e. x � r0f0, we write thewhistler wave dynamical equation in terms of beam widthparameter f0 with the help of Eqs. (12), and (14)–(16) as,

d2f0

dz2= (1 + ε−00

ε00)2

4f 30

− (1 + ε−00ε00

)α0(E0)2R2

d

2r20f 2

0

exp

(α0(E

0)2

f0

)(17)

where Rd = k‖r20 .

The first term on the right-hand side (RHS) of Eq. (17) isa diffraction term responsible for the divergence of the waveand the second nonlinear term is associated with the pon-deromotive nonlinearity that accounts for the convergence

of the wave. When both terms balance each other, the radialintensity distribution remains constant i.e. the wave propa-gates without convergence and divergence.

To elucidate the whistler wave localization due to lin-ear wave propagation and refraction in the inhomogeneousplasma, we solve Eq. (17) numerically by applying theboundary conditions f0 = 1 and df0

dz= 0 at z = 0. The basic

plasma parameters typically for Earth’s magnetosheath re-gion are: B0 = 8.2 × 10−4 G, n0 = 35 cm−3, Te = 3.2 ×105 K, r0 = 1.2 × 1010 cm, ω = 126 Hz, ωpe = 1.59 ×105 Hz, Rd = 7.19 × 1013 cm, k‖ = 4.99 × 10−7 cm−1.

3 Results and discussion

We use the fourth order Runge-Kutta method to solve thewhistler wave dynamical equation. The initial value of themagnetic field of whistler wave is assumed greater thanthe critical magnetic field for establishing the filamenta-tion instability. As a result, the converging force that arisesdue to the nonlinear term (the second term on RHS ofEq. (17)) starts dominating over the diverging force (diffrac-tion term, the first term on RHS of Eq. (17)). The phe-nomenon leads to the development of localized structuresof whistler wave.

3.1 Semi-analytical model results

Figure 1(a) depicts the variation of whistler beam width pa-rameter f0 in the direction of propagation i.e. z-axis (wherez is normalized by Rd ). Figure 1(b) manifests the magneticfield localized structures with the direction of propagation z.When the initial magnetic field of the whistler is greaterthan the critical magnetic field, the nonlinear term eventu-ally takes control over the diffraction term and the magni-tude of f0 dwindle in the direction of propagation. Afterreaching the critical minimum, the diffraction term startsdominating and the magnitude of f0 increases in the dis-tance of propagation. However, when the nonlinear term hassame magnitude as the diffraction term which governs bythe finite transverse size and wave number, whistler wavepropagates with a constant amplitude without convergingand diverging and we get a self trapping mode. At maxi-mum f0, once again nonlinear term starts dominating, andthe phenomenon repeats.

Figure 2(a) reveals more insight on the whistler wave lo-calized structure formation with a distance in the directionof propagation (z-axis), across x = 0 (where x is normal-ized by the transverse scale size r0 of the wave). Figure 2(b)shows the localized structures of the whistler magnetic fieldintensity (where B2 normalized by B2

0 ) at different loca-tions in xz-plane where x is normalized by the transversescale size r0 of the wave. The whistler wave attains a crit-ical minimum value f0 and the magnetic field intensity inthese small structures gets amplified (B2/B2

0 > 1) as shown

Nonlinear whistler wave model for lion roars in the Earth’s magnetosheath Page 5 of 7 172

Fig. 1 (a) Spatial distribution of beam width parameter f0 with the distance of propagation z (where z is normalized by Rd ). (b) Whistler wavelocalized structures with the distance of propagation z (where B2 is normalized by B2

0 )

Fig. 2 (a) Magnetic field localized structures of whistler wave across x (where x is normalized by r0). (b) Whistler wave localized structure inxz-plane (where B2 is normalized by B2

0 , x is normalized by r0, and z is normalized by Rd )

in Fig. 2a and b. In the present study, the fluctuations in thebackground field are assumed strong i.e. B2/B2

0 > 1 in con-trast to the weak turbulence limit where the nonlinearitiesare assumed small and fluctuations on the background areweak i.e. B2/B2

0 < 1 hence neglected.Figure 3 illustrates the whistler magnetic field power

spectrum at x = 0 and the energy distribution over the differ-ent wave numbers. The magnetic field localized structuresof whistler wave either lead to the decaying waves or actas a source for the further collapse of whistler wave. The

magnetic field spectrum and spectral index of the whistlerwill consequently change. We interpret the energy distribu-tion over the intermediate and high wave numbers due tothe nonlinear interaction between ion acoustic and whistlerwave modes. Moreover, the figure depicts a spectral breakafter which the spectrum becomes steeper with a spectraldependence of k−3.2 at higher wave numbers. Theoreticallyobtained energy spectrum demonstrates a similar behavioras the observed spectrum reported by Rodriguez (1985) athigher wave numbers. But no signature of the spectral peak

172 Page 6 of 7 N.K. Dwivedi, S. Singh

Fig. 3 Magnetic field power spectrum against k at x = 0 (where x isnormalized by r0, and k is normalized by R−1

d )

in the simulated energy spectrum (at a larger scale) is ob-tained as it is observed by Rodriguez (1985).

3.2 Comparison with the observational results

Lion roars are observed in the magnetosheath (Baumjohannet al. 1999) region and posses similar characteristics as thewhistler wave found in the dawn side equatorial magneto-sphere (Baumjohann et al. 2000). Rodriguez (1985) inves-tigated the short-period magnetic field fluctuations in theEarth’s magnetosheath and reported broadband turbulencespectrum with a spectral index of −4.5. A narrow-bandfrequency peak is superimposed on the broadband spec-trum, which is a realization of short period lion roars bursts.Note that the observed broadband turbulence spectrum is inthe frequency domain, whilst, the whistler wave turbulencespectrum estimated in the present work using the frame oftwo-fluid approximation is in the wave number domain. Toperform a direct comparison between the simulated turbu-lence spectrum and the observational spectrum (Rodriguez1985), the spectral slope needs to be derived in the frequencydomain. We propose an analytical scenario and utilize thedispersion relation of whistler wave propagating along themagnetic field to obtain the spectral slope value in the fre-quency domain.

Dispersion relation of whistler wave is given as (Biskampet al. 1996),(

ω

Ωe

)2

=(

kc

ωpe

)2[1 + k2c2cos2θ

ωpe2

](18)

where ω is the frequency of the whistler wave, Ωe is elec-tron gyro-frequency, ωpe electron plasma frequency, k wavenumber, and c is the light speed. For parallel propagatingwhistler wave, we can simplify the dispersion relation as,

ω

Ωe

= kk‖c2

ωpe2

(19)

where k‖ denotes the parallel wave number.

Fig. 4 Schematic diagram of the magnetic field turbulence spectrumof lion roars (observational), the broadband spectrum of whistler wave(observational), and whistler wave spectrum (theoretical)

We use the dispersion relation to compute magnetic fieldspectrum in the frequency domain,

ξ(ω) =∫ ∞

−∞Ck−α

‖ δ(ω − k−2

‖)dk‖ (20)

ξ(ω) = Cω−( α+12 ) (21)

where C is a constant, and α is the spectral slope value.We find the spectral slope of the theoretical whistler tur-

bulence spectrum −5.4, which is relatively steeper than theobservational broadband spectrum of spectral slope −4.5.Figure 4 shows a schematic illustration of the short periodlion roars turbulence spectrum, the observational whistlerwave broadband spectrum, and the theoretically predictedwhistler wave turbulence spectrum in the frequency do-main. We utilize this plot to highlight the main observationsthat emerge from the comparison between the observationaland theoretical spectral slopes. Note that this illustration ismerely a schematic plot with arbitrary tick labels. It can benoticed that the spectral knee is absent from the both, obser-vational and theoretical, whistler wave turbulence spectrum,unlike the lion roars turbulence spectrum, which clearly in-dicates a sharp peak at a lower frequency. The present studyinfers that the two-fluid whistler wave model can reason-ably explain the magnetic field spectra and the correspond-ing spectral slope values as observed by IMP6 plasma wavesand magnetometer experiment (Rodriguez 1985).

4 Conclusion

We present a two-fluid steady state model to investigate theturbulence properties in Earth’s magnetosheath due to thenonlinear spatial evolution of whistler waves. The energyspectrum obtained from the present model shows a spec-tral break and becomes steeper at a higher wave number

Nonlinear whistler wave model for lion roars in the Earth’s magnetosheath Page 7 of 7 172

side with a spectral slope of −3.2. The semi-analytical re-sults provide important physical insights. The localizationand delocalization of the wave in the direction of propaga-tion imply that undergoing filamentation instability leads tothe magnetosheath plasma turbulence. In addition, compar-ison of the theoretical model results with the observationaldata provides a good evidence in support of the present two-fluid model to investigate steady state self-focusing whistlerwaves. Nevertheless, it is imperative to investigate the mag-netic field fluctuations and turbulence evolution with timefor a better prediction of the magnetosheath turbulence prop-erties. Based on the present study, research work focusing ontransient turbulence effects is under way and will be reportedseparately in the near future.

Acknowledgements Authors like to thank Prof. R.P. Sharma for hisvaluable suggestions.

References

Akhiezer, A.I., Akhiezer, A.I., Polovin, R.V., Sitenko, A.G., Stepanov,K.N.: Plasma Electrodynamics. Vol. 2. Nonlinear Theory andFluctuations. Pergamon, New York (1975)

Baumjohann, W., Treumann, R.A., Georgescu, E., Haerendel, G., For-nacon, K.-H., Auster, U.: Ann. Geophys. 17, 1528–1534 (1999)

Baumjohann, W., Georgescu, E., Fornacon, K.-H., Auster, H.U.,Treumann, R.A., Haerendel, G.: Ann. Geophys. 18, 406–410(2000)

Bhattacharjee, A., Ng, C.S., Spangler, S.R.: Astrophys. J. 494, 409(1998)

Biskamp, D., Schwarz, E., Drake, J.F.: Phys. Rev. Lett. 76, 1264(1996)

Burman, R.R.: Publ. Astron. Soc. Jpn. 27, 511 (1975)Cornilleau-Wehrlin, N., Chauveau, P., Louis, S., Meyer, A., Nappa,

J.M., Perraut, S., Rezeau, L., Robert, P., Roux, A., de Velledary,C., de Conchy, Y., Friel, L., Harvey, C.C., Hubert, D., Lacombe,C., Manning, R., Wouters, F., Lefeuvre, F., Parrot, M., Pincon,J.L., Poirier, B., Kofman, W., Louarn, P.: Space Sci. Rev. 79, 107–136 (1997)

Davidson, R.C.: Methods in Nonlinear Plasma Theory. Academic, NewYork (1972)

Dwivedi, N.K., Sharma, R.P.: Phys. Plasmas 20, 042308 (2013)Dwivedi, N.K., Batra, K., Sharma, R.P.: J. Geophys. Res. 117, A07201

(2012)Gary, S.P., Saito, S., Li, H.: Geophys. Res. Lett. 35(2), L02104 (2008)Goldstein, B.E., Tsurutani, B.T.: J. Geophys. Res. 89, 2789–2810

(1984)

Hasegawa, A.: Plasma Instabilities and Nonlinear Effects. Springer,New York (1975)

Kadomtsev, B.B.: Plasma Turbulence. Academic, New York (1965)Krafft, C., Volokitin, A.: Ann. Geophys. 21, 1393 (2003)Maksimovic, M., Santolik, C.C., Lacombe, C., de Conchy, Y., Hubert,

D., Pantellini, F., Cornilleau-Werhlin, N., Dandouras, I., Lucek,E.A., Balogh, A.: Ann. Geophys. 19, 1429–1438 (2001)

Masood, W., Schwartz, S.J., Maksimovic, M., Fazakerley, A.N.: Ann.Geophys. 24, 1725–1735 (2006)

Ng, C.S., Bhattacharjee, A., Germaschewiski, K., Galtier, S.: Phys.Plasmas 10, 1954 (2003)

Nishida, A.: Geophys. Res. Lett. 21, 2871–2873 (1994)Palmadesso, P., Papadopoulos, K.: Waves and Instabilities in Space

Plasmas. Reidel, Dordrecht (1979)Qureshi, M.N.S., Nasir, W., Masood, W., Yoon, P.H., Shah, H.A.,

Schwartz: J. Geophys. Res. 119, 10,059–10,067 (2014)Rodriguez, P.: J. Geophys. Res. 90, 241–248 (1985)Roth, I.: Planet. Space Sci. 55, 2319 (2007)Sagdeev, R.Z., Galeev, A.A.: Nonlinear Plasma Theory. Benjamin,

New York (1969)Saito, S., Gary, S.P., Li, H., Narita, Y.: Phys. Plasmas 15, 102305

(2008)Santolík, O., Gurnett, D.A., Pickett, J.S., Parrot, M., Cornilleau–

Wehrlin, N.: J. Geophys. Res. 108, 1278 (2003)Sharma, R.P., Goldstein, M.L., Dwivedi, N.K., Chauhan, P.K.: J. Geo-

phys. Res. 115, A12207 (2010)Smith, E.J., Tsurutani, B.T.: J. Geophys. Res. 81, 2261–2266 (1976)Smith, E.J., Holzer, R.E., McLeod, M.G., Russel, C.T.: J. Geophys.

Res. 72, 4803 (1967)Smith, E.J., Holzer, R.E., Russel, C.T.: J. Geophys. Res. 74, 3027

(1969)Stawicki, O., Gary, S.P., Li, H.: J. Geophys. Res. 106, 8273 (2005)Thorne, R.M., Tsurutani, B.T.: Nature 293, 384–386 (1981)Tsurutani, B.T., Smith, E.J., Anderson, R.R., Ogilvie, K.W., Scudder,

J.D., Baker, D.N., Bame, S.J.: J. Geophys. Res. 87, 6060–6072(1982)

Tsurutani, B.T., Verkhoglyadov, O.P., Lakhina, G.S., Yagitani, S.: J.Geophys. Res. 114, A03207 (2009)

Tsurutani, B.T., Falkowski, B.J., Verkhoglyadov, O.P., Pickett, J.S.,Santolík, O., Lakhina, G.S.: J. Geophys. Res. 116, A09210 (2011)

Vedenov, A.A.: Theory of Turbulent Plasma. Elsevier, New York(1968)

Vocks, C., Salem, C., Lin, R.P., Mann, G.: Astrophys. J. 627, 540(2005)

Wei, X.H., Cao, J.B., Zhou, G.C., et al.: J. Geophys. Res. 112, A10225(2007)

Yoon, P.H.: Phys. Plasmas 7, 4858 (2000)Yoon, P.H., Ziebell, L.F., Gaelzer, R., Pavan, J.: Phys. Plasmas 19,

102303 (2012)Yoon, P.H., Ziebell, L.F., Kontar, E.P., Schlickeiser, R.: Phys. Rev. E

93, 033203 (2016)Zhang, Y., Matsumoto, H., Kojima, H.: J. Geophys. Res. 103, 4615–

4626 (1998)