Modulational instability of electromagnetic electron-cyclotron wave packets in themagnetosphere with non-Maxwellian electron distributionH. Abbasi and H. Hakimi Pajouh

Citation: Physics of Plasmas (1994-present) 15, 092902 (2008); doi: 10.1063/1.2978192 View online: http://dx.doi.org/10.1063/1.2978192 View Table of Contents: http://scitation.aip.org/content/aip/journal/pop/15/9?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Resonance zones for electron interaction with plasma waves in the Earth’s dipole magnetosphere. II. Evaluationfor oblique chorus, hiss, electromagnetic ion cyclotron waves, and magnetosonic waves Phys. Plasmas 17, 042903 (2010); 10.1063/1.3310835 Resonance zones for electron interaction with plasma waves in the Earth’s dipole magnetosphere. I. Evaluationfor field-aligned chorus, hiss, and electromagnetic ion cyclotron waves Phys. Plasmas 17, 042902 (2010); 10.1063/1.3310834 Relativistic modulational instability of electron-acoustic waves in an electron-pair ion plasma Phys. Plasmas 15, 122107 (2008); 10.1063/1.3050062 Dynamics of nonlinearly coupled magnetic-field-aligned electromagnetic electron-cyclotron waves near the zero-group-dispersion point in magnetized plasmas Phys. Plasmas 12, 082303 (2005); 10.1063/1.1994747 Modulation of electromagnetic electron cyclotron waves in the presence of nonisothermal electrons in plasmas Phys. Plasmas 11, 4346 (2004); 10.1063/1.1774165

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Modulational instability of electromagnetic electron-cyclotron wave packetsin the magnetosphere with non-Maxwellian electron distribution

H. Abbasi1,a� and H. Hakimi Pajouh2

1Faculty of Physics, Amirkabir University of Technology, P.O. Box 15875-4413, Tehran, Iran2Department of Physics, Alzahra University, Tehran 19834, Iran

�Received 8 July 2008; accepted 15 August 2008; published online 17 September 2008�

The satellite observations of the magnetosphere in the low-frequency ion dynamics revealed severalfacts: �1� localized structures for electric field signal parallel to the magnetic field; �2� anisotropy forthe electron velocity distribution such that T��T�; and �3� non-Maxwellian distribution function forthe electrons. Based on these evidences, a nonlinear model is presented to develop theelectromagnetic electron-cyclotron �EMEC� theory to the non-Maxwellian plasmas. Then, themodulational instability analysis of EMEC waves is investigated. © 2008 American Institute ofPhysics. �DOI: 10.1063/1.2978192�

I. INTRODUCTION

Among different types of cyclotron waves in magneto-active plasma,1–5 electromagnetic electron-cyclotron�EMEC� waves are frequently observed in the Earth’smagnetosphere.6,7 They might be generated through a varietyof linear instabilities, such as the one associated with elec-tron temperature anisotropy or streaming instabilities. Longwavelength EMEC waves can be excited by a lightningstrike. The resulting electromagnetic energy can be guidedalong magnetic field lines. Since the magnetosphere is a dis-persive medium, different frequencies propagate with differ-ent phase velocities resulting in descending whistles that canbe heard on a receiver, over the radio frequency range, afterlightning. The modulated EMEC wave packet associatedwith quasistationary density profiles have been detected byrecent satellite observations8,9 in the magnetosphere and inlaboratory experiments.10–13

Generally, for the large amplitude EMEC waves, nonlin-ear effects become significant. Among the nonlinear effectsin dispersive media, is the nonlinear modulation of high-frequency EMEC wave through its coupling with low-frequency perturbations. Following the early work ofHasegawa,14,15 who used a reductive perturbation method tostudy the self-modulation of EMEC waves, Karpman andWashimi16 included the ponderomotive force in the formal-ism by which the modulation of EMEC waves by low-frequency perturbations could be studied. Recently, the for-malism was adopted as a model for the localized whistler-related envelope structures coupled to densityperturbations,17,18 which are frequently observed in theEarth’s magnetosphere.

This paper is an attempt to develop, based on the PolarOrbiting Satellite �POLAR� �Refs. 19 and 20� and Fast Au-roral Snapshot �FAST� �Refs. 21 and 22� satelliteobservations of the magnetosphere, the EMEC theory to

non-Maxwellian plasmas. The satellite observations of themagnetosphere in the low-frequency ion dynamics revealedseveral facts: �1� localized structures for electric field signalparallel to the magnetic field;21 �2� anisotropy for the elec-tron velocity distribution such that T��T�;

22 and �3� non-Maxwellian distribution function �DF� for the electrons.22

For this purpose, a kappa or generalized Lorentzian DF, de-noted by �, is introduced to model the non-Maxwellian DFof electrons. It is because, in space physics that energeticparticles are ubiquitously observed,23–29 they are often mod-eled by the � distribution.23,30,31 It is widely believed that theorigin of the energetic particles lies in the acceleration bywave turbulence, i.e., second-order Fermi acceleration.

Besides, in some observations the velocity of the local-ized structures has been reported to be substantially greaterthan the ion acoustic speed. In a Maxwellian plasma, thisfact is attributed to the formation of Bernstein–Green–Kruskal electron holes which are a highly nonlinear state.Their theoretical explanation32,33 involves a depression in theelectron distribution function, supported by the populationsof trapped and free particles in phase space. Most of thetheoretical treatments that have been done, so far, assumes aMaxwellian DF for the electrons.22,34–39

Here, we present a nonlinear formalism emphasizing thenon-Maxwellianity of the electron DF �and DF-related phe-nomena such as the electron trapping� when the electrontemperature is anisotropic. We derive the governing equa-tions for the modulated EMEC waves taking into accountrelativistic electron mass increase as well as electron densitychanges in the presence of the radiation pressure modified bytrapped electrons. The governing equations are then used toinvestigate the influence of electron trapping and non-Maxwellianity of electron DF on the modulational instabilityof the EMEC field. The plasma is assumed to be collisionlessand unbounded with cold ions.

The manuscript is organized as follows: In Sec. II, anenvelope equation for the amplitude of EMEC is obtained. InSec. III, this equation is solved for the study of the modula-tional instability. Section IV contains the conclusion.a�Electronic mail: abbasi@theory.ipm.ac.ir.

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II. ENVELOPE EQUATION

We consider the nonlinear propagation of circularly po-larized EMEC waves in a constant external magnetic fieldB0=B0ez, where B0 is the strength of the constant magneticfield and ez is the unit vector along the z axis. The waveelectric field is

E� = 12 �ex + iey�E��z,t�e−i�t + c.c., �1�

where ex �ey� is the unit vector along the x �y� axis, � is thewave frequency, and c.c. stands for the complex conjugate.The “�” sign denotes the perpendicular direction to the B0.

There are two distinguishable time scales in our prob-lem: One for the fast motion with the characteristic time ofthe order of the high-frequency �hf� EMEC wave, �=2� /�,and another for the low-frequency �lf� perturbations. There-fore, it is reasonable to represent any quantity in the form ofa sum which consists of the slowly and the rapidly varying

parts, A= �A�+ A, where the angular bracket denotes averag-ing over the time interval �.

From Maxwell equations we can derive the wave equa-tion governing the fast varying part of the electric field as

�2E� −1

c2

�2

�t2 E� +4�e

c2 �ne��

�tve� = 0, �2�

where �ne� and ve are the slowly varying density and therapidly varying velocity of the electrons, respectively, e isthe magnitude of the electron charge, and c is the speed oflight in vacuum. In the following, we shall focus on theone-dimensional problem. That means the parallel spatialscale is assumed to be much smaller than the perpendicularextension of the electromagnetic wave packet. In this way, it

is reasonable to assume �= k� /�z.Fast motion of the electrons can be described by the

momentum equation

�

�tpe� = − eE� −

e

c�ve� � B0� , �3�

where the relativistic momentum is

pe� =meve�

�1 − ve�2 /c2

. �4�

Here, me is the electron mass. Equation �4� can be solved byintroducing

ve� = 12 �ex + iey�ve��z,t�e−i�t + c.c., �5�

and assuming first order of smallness for E� �the amplitudeof the hf field in Eq. �1�� and second order of smallness forthe temporal derivative of the slowly varying amplitudes, inthe following form:

ve� = −ie

me�� − �c�E� −

e

me�� − �c�2

�E�

�t

+ ie3

2me3c2

�

�� − �c�4 E�2E�, �6�

where �ceB0 /mec. The last term in Eq. �6� is associatedwith the relativistic electron mass variation.

Substituting Eqs. �1�, �5�, and �6� into Eq. �2� and assum-ing that the modulation frequency is much smaller than thecarrier one, we obtain

2i��1 +1

2

�p2

�2

��c

�� − �c�2� �E�

�t+ c2�2E�

�z2 + �2E�

− � ��p2

� − �c−

e2

2me2c2

�2�p2

�� − �c�4 E�2� �ne�n0

E� = 0,

�7�

where n0 is the equilibrium electron density and �p

= �4�e2n0 /me�1/2 is the electron plasma frequency.In order to define the slowly varying part of the electron

density, �ne�, taking into account the electron trapping in aplasma with non-Maxwellian velocity distribution for theelectrons, we have to proceed from a kinetic description ofthe electron gas. As it was mentioned, the � DF is our choice.In the limit of �→�, the � DF approaches the Maxwellianone. The idea behind the model DF, containing both the freeand trapped electrons, is acquired by the result of the long-time simulation of Vlasov–Poisson system of equations�such as Ref. 40�. A comprehensive discussion in this respectis in Ref. 41, which results in the following expression forthe electron DF:

f t�ve�,ve�� =n0C�

vTe�2�

�1 + �ve�

2 − 2Ue/me

vTe2 �2� − 3� �−��ve��

2�ve�

,

ve� �2Ueme

, �8�

f f�ve�,ve�� =n0C�

vTe�2�

�1 +ve�

2 − 2Ue/me

vTe2 �2� − 3� �−��ve��

2�ve�

,

ve� ��2Ueme

, �9�

where f f and f t are the free and trapped velocity DF, respec-tively. is the Dirac delta function and is used to model theobserved electron temperature anisotropy, Te��Te�. Te� is

Te� =me

3lim�→�

0

�

2�ve�dve� −�

+�

ve�2 f fUe=0 dve� ,

vTe is the electron thermal velocity ��Te� /me�, � is the spec-tral index ���3 /2�, and � is the trapping parameter. �=0models the plateau structure and � 0 models the hole struc-ture in the electron distribution function. C� is the normal-ization factor as follows:

C� =1

�� − 3/2����

��� − 1/2�, �10�

where � is the gamma function. Ue is the averaged electronpotential energy. It is composed of two parts,

Ue = − e� +e2

2me��� − �c�E2. �11�

The first term is the charge separation potential energy andthe second one is the ponderomotive potential. Since the

092902-2 H. Abbasi and H. Hakimi Pajouh Phys. Plasmas 15, 092902 �2008�

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present model contains the electron trapping, Ue is assumedto have the form of a single potential well, vanishing atinfinity.

The electron density is then obtained by integrating theDF over the corresponding velocity range �the trapped andfree electrons occupy distinct range of the velocity space�.The result is the following expression for the density:

�ne�n0

=1

��

������� + 1/2��2F1�1

2,� −

1

2;� +

1

2;1 −

Ue/Te�

�� − 3/2��+

2� − 1�� − 3/2

�Ue/Te�2F1��,1;3

2;�Ue/Te�

� − 3/2 �� , �12�

where 2F1 is the hypergeometric function,

2F1�i, j ;k;z� =��k�

��j���k − j� 0

1

dttk−1�1 − t�k−j−1�1 − tz�−i,

Rk � Rj � 0. �13�

In the weak nonlinear regime �Ue /Te� 1�, we expandEq. �12� up to the first nonlinear term. That is,

�ne�n0

= 1 + a�

UeTe�

−4

3

�1 − ����

b�� UeTe�

�3/2

, �14�

where

a� =2� − 1

2� − 3, �15�

b� =��� + 1�

�� − 3/2�3/2��� − 1/2�. �16�

The ions participate in the slowly varying dynamics ofdriven ion-acoustic perturbations involving the ion numberdensity ni�ni� and the ion fluid velocity vi�vi�. The ap-propriate ion and Poisson’s equations for our purposes aregoverned by

�ni

�t+

�

�z�nivi� = 0, �17�

�vi

�t+ vi

�vi

�z= −

e

mi

��

�z, �18�

�2�

�z2 = 4�e��ne� − ni� , �19�

where mi is the ion mass. Neglecting nonlinear terms in Eqs.�17� and �18� we can combine them to obtain ni for the caseof stationary ion-acoustic waves with a constant velocity, M.The result is

ni

n0= 1 +

cs2

M2

e�

Te�

, �20�

where cs= �Te� /mi�1/2 is the ion-acoustic speed.As the FAST observations showed, in some cases, the

parallel size of the electrostatic structures is longer than thelocal plasma Debye radius.21,22 Therefore, the condition forthe quasineutrality is considered, that is, �ne���ni�. In the

other cases that the parallel size is comparable with the localplasma Debye radius, the following formalism cannot be ap-plied. Accordingly, Eqs. �11�, �14�, and �20� can be solvedfor the slowly varying electron density as

�ne�n0

= 1 +e2

2meTe�

cs2

M2 − cs2/a�

1

��� − �c�E�2

+4�1 − ��

3��� e2

2meTe��3/2 b�

a�5/2

cs2

M2 − cs2/a�

� � cs2

M2 − cs2/a�

1

��� − �c��3/2

E�3. �21�

Substituting Eq. �21� into Eq. �7�, we obtain a modified non-linear Schrödinger equation,

2i��E�

�t+ c2�2E�

�z2 + ��2 −��p

2

� − �c�E� + CfE�2E�

− CtE�3E� = 0, �22�

� = ��1 +1

2

�p2

�2

��c

�� − �c�2� , �23�

Cf =e2

2meTe�

�p2

�� − �c�2�vTe2

c2

�2

�� − �c�2 −cs

2

M2 − cs2/a�

� , �24�

Ct =4�1 − ��

3��� e2

2meTe��3/2

�b�

a�5/2

��p2

� − �c

cs2

M2 − cs2/a�

� cs2

M2 − cs2/a�

1

��� − �c��3/2

,

�25�

where Cf and Ct are the coefficients of nonlinear terms asso-ciated with the free and trapped electrons, respectively. In theframework of the above equation, the modulational instabil-ity of the EMEC are studied in the next section.

III. MODULATIONAL INSTABILITY ANALYSIS

In this section, the modulational instability analysis ofEq. �22� is presented. Accordingly, we let

E� = ��0 + ��z,t��ei�kz−�t�, �26�

where �0 �real� is the amplitude of the transversal wave,����0� is the complex amplitude of the perturbation, � is anonlinear frequency shift produced by the nonlinear interac-tion, and k is the wave number.

On substituting Eq. �26� into Eq. �22� and consideringthe terms corresponding to the zero order of �, we obtainthe nonlinear frequency shift

� =1

2�� ��p

2

� − �c− �2 + k2c2 − Cf�0

2 + Ct�03� . �27�

By using Eq. �27�, we can write the equation governingon the perturbation amplitude as

092902-3 Modulational instability of electromagnetic… Phys. Plasmas 15, 092902 �2008�

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2i�� ��

�t+ vg

��

�z� + c2�2�

�z2 + �Cf�02 −

3

2Ct�0

3�R� = 0,

�28�

where vg=kc2 /� and R�=�+�*.A combination of Eq. �28� with its complex conjugate

leads to

2i�� �R�

�t+ vg

�R�

�z� + c2�2I�

�z2 = 0, �29�

2i�� �I�

�t+ vg

�I�

�z� + c2�2R�

�z2 + 2�Cf�02 −

3

2Ct�0

3�R� = 0,

�30�

where I�=�−�*. The dispersion relation for the modula-tional instability can now be derived by letting

�R�

I�� = �R�

I�

�ei��z−�mt�, M =�m

�, �31�

where � and �m are the modulation wavelength and fre-

quency, respectively. R� and I� are real. The resulting equa-tions can be combined to yield

� 2���m − vg�� − c2�2

2Cf�02 − 3Ct�0

3 − c2�2 2���m − vg�� ��R�

I�� = 0.

�32�

Then, the dispersion relation is obtained as

��m − vg��2 +c2�2

4�2 �2Cf�02 − 3Ct�0

3 − c2�2� = 0, �33�

where the new definition of Cf and Ct are the following:

Cf =e2

2meTe�

�p2

�� − �c�2�vTe2

c2

�2

�� − �c�2 −cs

2�2

�m2 − cs

2�2/a�� ,

�34�

Ct =4�1 − ��

3��� e2

2meTe��3/2 b�

a�5/2

��p2

� − �c

�cs

2�2

�m2 − cs

2�2/a�� cs

2�2

�m2 − cs

2�2/a�

1

��� − �c��3/2

. �35�

Let us now consider the validity of Eq. �26�. For thispurpose, suppose that the characteristic temporal and spacialscales of the nonlinear background �EMEC wave packet� are� and L, respectively. Then, in order to ignore the spatial andtemporal dependencies of the wave packet, these scales haveto fulfill the inequalities 1 /L���k and 1 /���m��. Fol-lowing these criteria, the corresponding envelope variationare negligible and the envelope might be considered similarto the case where the background field is uniform ��0

=constant�. Moreover, to observe only the temporal growthrate, the phase velocity of the modulated wave have to beequal to the group velocity of the EMEC wave packet, that

is, �m=�vg+ i�����vg�, where i is the imaginary unit.Then, upon substituting �m into Eq. �33�, the growth rate ofthe modulated waves is obtained as follows:

� =1

2

c

���2Cf�0

2 − 3Ct�03 − c2�2. �36�

In what follows, we show how the growth rate depends ontwo parameters, � and �, corresponding to the electron trap-ping and non-Maxwellianity of the electron DF, respectively.The typical plasma parameters of the Earth’s magnetosphericelectron-ion �hydrogen ions� that has been used in the nu-merical evaluation of � are n0=6 cm−3, Te� =750 eV, �=75 kHz, and B0�410 nT. The electric field amplitude, �0,is assumed to be 1 mV /cm.

Figure 1 shows the variation of the nonlinear frequencyshift, �, against �. The result demonstrates a monotonicallyincreasing behavior of nonlinear frequency respect to �.

Figure 2 exhibits the variation of the growth rate versusthe wave number of the modulated waves for fixed �=0. Thestabilization effect regarding the non-Maxwellian DF ofelectrons can be seen from significant reduction of thegrowth rates. From the figure, one can deduce that the rangeof unstable � becomes smaller for the smaller �. Further-more, for two different �, the growth rate of a fixed �, issmaller for the smaller �. In the other words, as the deviationof electron DF from the Maxwellian profile becomes larger,the modulational instability is weakened. In conclusion, thelargest instability range �� range� and growth rate belongs tothe Maxwellian plasma.

The role of the electron hole depth, �, on � is demon-strated in Fig. 3. The fixed parameter is �=2. It signifies thatas the electron hole becomes deeper �more negative ��, thegrowth rate decreases. In the other words, for the deeperelectron hole the condition for the modulational instabilitybecomes harder and the region of unstable � becomessmaller and is restricted to larger wavelengths.

κ

∆

2 20 38 56 74 922962

2985

3008

3031

3054

(s)

-1

FIG. 1. � vs � for fixed �=0.

092902-4 H. Abbasi and H. Hakimi Pajouh Phys. Plasmas 15, 092902 �2008�

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IV. CONCLUSIONS

Based on the POLAR and FAST satellite observations ofthe magnetosphere in the low-frequency ion dynamics, anonlinear model was presented with the following properties:�I� non-Maxwellian electron DF; �II� anisotropic electron DFthat means T��T�; �III� relativistic electron mass variation;�IV� electron trapping; and �V� cold ions. In this framework,a modified nonlinear Schrödinger equation was obtained bywhich the modulational instability of the EMEC waves wasstudied. It was shown that deviation from the Maxwellianequilibrium significantly reduces the growth rate of the in-stability. Moreover, for the larger deviation �smaller ��, the

wavelength range over which the instability takes place isrestricted to the larger wavelength. The effect of electrontrapping can be explained by the electron-hole depth, in thisway that by increasing the depth of electron-hole, the growthrate decreases and the wavelength range of instability is lim-ited to the larger wavelength.

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χ

γ

0 0.01 0.02 0.03 0.04 0.050

0.1

0.2

0.3

0.4

0.5

0.6

(Km )-1

(s)

-1

κ = 2

κ = 2.2

κ = 2.6

κ = 4

Maxwellian

FIG. 2. � vs � for fixed �=0 and B0=410 nT.

χ

γ

0 0.001 0.002 0.003 0.004 0.005 0.006 0.0070

0.004

0.008

0.012

= 0.0= -0.5= -1.0

(Km )-1

(s)

-1

β

ββ

FIG. 3. � vs � for fixed �=2 and B0=410 nT.

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092902-6 H. Abbasi and H. Hakimi Pajouh Phys. Plasmas 15, 092902 �2008�

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