Magnetic electron drift waves in electron magnetohydrodynamic plasmasNikhil Chakrabarti and Raghvendra Singh Citation: Physics of Plasmas (1994-present) 11, 5475 (2004); doi: 10.1063/1.1815002 View online: http://dx.doi.org/10.1063/1.1815002 View Table of Contents: http://scitation.aip.org/content/aip/journal/pop/11/12?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Impact of resonant magnetic perturbations on nonlinearly driven modes in drift-wave turbulencea) Phys. Plasmas 19, 055903 (2012); 10.1063/1.3694675 Coupled nonlinear drift and ion acoustic waves in dense dissipative electron-positron-ion magnetoplasmas Phys. Plasmas 16, 112302 (2009); 10.1063/1.3253623 Magnetodynamics of a multicomponent (dusty) plasma. II. Magnetic drift waves in an inhomogeneous medium Phys. Plasmas 12, 042111 (2005); 10.1063/1.1881513 Sheared-flow-driven ion-acoustic drift-wave instability and the formation of quadrupolar vortices in a nonuniformelectron–positron–ion magnetoplasma Phys. Plasmas 11, 4341 (2004); 10.1063/1.1774164 Magnetic-curvature-driven interchange modes in dusty plasmas Phys. Plasmas 11, 542 (2004); 10.1063/1.1640621

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Magnetic electron drift waves in electron magnetohydrodynamic plasmasNikhil ChakrabartiSaha Institute of Nuclear Physics, 1/AF Bidhannagar, Calcutta 700 064, India

Raghvendra SinghInstitute for Plasma Research, Bhat, Gandhinagar 382428, India

(Received 2 June 2004; accepted 21 September 2004; published online 1 November 2004)

The dynamics of nonuniform, magnetized, cold electron plasma in a stationary charge neutral ionbackground is considered. In the high frequency limit, electromagnetic modification of electronplasma oscillation is obtained in homogeneous plasma, whereas in inhomogeneous plasma in thelow frequency regime a driftlike mode is found. Nonlinear evolution of this mode is derived todescribe the two-dimensional(2D) dynamics. The equation has a close resemblance to 2Delectrostatic collisionless drift wave equation thereby called as “magnetic electron drift” wave. Inaddition to the usual small amplitude dispersive modes the stationary state nonlinear vortexlikesolutions are also discussed. The magnetic electron drift wave in the electron magnetohydrodynamicregime can find applications in laboratory as well as in space plasmas. ©2004 American Instituteof Physics. [DOI: 10.1063/1.1815002]

I. INTRODUCTION

The propagation of finite amplitude waves in a low den-sity magnetized plasma exhibits a number of distinguishedfeatures in both the linear and nonlinear regimes. This isunique feature of magnetofluids(plasmas) as compared toordinary fluids. In plasmas the entire wave propagations arebased on mainly two models, viz.(i) magnetohydrodynam-ics, (ii ) electron-magnetohydrodynamics(EMHD). The firstone is very common and widely used for the phenomenawhich occurs in a slow time scale. In this case, ion dynamicsplays an important role, while electrons respond very quicklyto shielding the instantaneous charge imbalance. The secondmodel, i.e., EMHD, has been used in the past and recentlyhas become popular. It is mainly used to describe the highfrequency phenomena, i.e., when the relevant time scales arefast. In this model electrons take part in the dynamics,whereas the ions provide a charge neutralized static back-ground. Here we are interested in the waves in EMHDmodel.

The theory of EMHD waves has been extensively stud-ied in a wide range of physical contexts starting from labo-ratory experiments1 to astrophysics,2 space physics, solarphysics, etc. In spite of a long history of investigations, thereare many important questions which still attract attention todifferent interesting physical phenomena, such as magneticturbulence,3 fast magnetic field penetration in plasma open-ing switches,4 reconnection of magnetic field lines,5,6 andmany other problems.7 In this paper, we have studied thepossibility of linear and nonlinear wave propagation in aninhomogeneous electron plasma. First we consider an elec-tron plasma with ion background where the density fluctua-tions are also finite. In such a case we find the modificationof simple electron plasma oscillation by electromagnetic ef-fect, namely, by electron magnetosonic wavev2=vp

2

+k2VAe2 , [where k is the perpendicular wave vector and

VAes=B0/4pmn00d is the electron Alfven speed,n00 is thedensity,m is the mass of the electron, andB0 is the equilib-

rium magnetic field]. This wave corresponds to the high fre-quency limit fv.vp=s4pn00e

2/md1/2g in a homogeneousplasma background. On the other hand, in the low frequencyregime fv,kc,vp,vcs=eB0/mcdg, we find a driftlike wavewith dispersion relationv=kydeVAe/Lns1+k2de

2d, whereLn isthe density scale length. Next, we have identified exactly thesame wave due to a density inhomogeneity in the EMHDmodel, where= ·J=0 condition ensures the absence of den-sity fluctuations. The origin of the wave is due to the com-pressibility condition of electron velocity. In an inhomoge-neous plasma where the density gradient is very sharp, theplasma tends to be uniform in that area. During this processthere are some charge fluctuations which might have set upsuch a wave in the plasma. The wave propagates perpendicu-lar to both the magnetic field as well as the direction ofinhomogeneity. The dispersion relation for this wave is onceagainv=kyvmd/1+k'

2 de2, wherevmds=devAe/Lnd is the mag-

netic drift velocity and the dispersion scalede is the electronskin depthsc/vpd. The wave frequency is lower than boththe plasma frequency and electron cyclotron frequencysv!vp,vcd and the scale length is larger than the electron skindepthsk'de!1d. The linear dispersion relation of this wavehas a close relation to that of the electrostatic drift wave.This relation is illustrated in consecutive sections. Not onlythe linear dispersion relation, but also the nonlinear evolutionresembles the well-known Hasegawa–Mima8 equation wherethe electrostatic potential fluctuations are replaced by mag-netic fluctuations. In the past an interesting work on highfrequency magnetic drift wave was done by Huba9 where theion dynamics and the Hall term are included. In the presentmodel ions do not at all take part in the dynamics. We hopethat a number of interesting analyses can be done with thispresent model in the EMHD regime, especially in the plasmaopening switches as well as in the EMHD turbulence.10 Inthe present paper we restrict ourselves only to a linear waveand a nonlinear vortexlike solution of the model describedhere.

PHYSICS OF PLASMAS VOLUME 11, NUMBER 12 DECEMBER 2004

1070-664X/2004/11(12)/5475/5/$22.00 © 2004 American Institute of Physics5475

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The paper is organized as follows. In Sec. II, we derivethe model equation starting from the basic set of equations,and in Sec. III, the linear dispersion relation is obtained andits relation with electrostatic drift waves is illustrated. In Sec.IV, the possible nonlinear vortexlike solution is derived. Weconclude with the summary of our work in Sec. V.

II. MODEL EQUATIONS AND LINEAR WAVES

In this section we consider the magnetic electron modemodel in an inhomogeneous plasma. The model neglects alleffects of magnetic geometry, considering only one fluctuat-ing variable, the axial magnetic field. The ions are assumedto be cold and fixed. The system described below is twodimensional(2D) (slab geometry) (i.e., ] /]z=0), no varia-tion along thez direction). The nonlinear equations are de-rived, assuming the time scale of the phenomena to be veryfast so that only electrons can participate, and the ions beingheavy, they form a static charge neutralizing background.The spatial scale lies below the ion inertial scalec/vpi. Sincewe restrict ourselves within 2D configurations, all the vari-ables are functions of the two spatial coordinatessx,yd andtime t only. For simplicity we assume the plasma to be col-lisionless. We start from the electron momentum equation

mnS ]

]t+ ve · = Dve = − neSE +

1

cve 3 BD + mnme¹

2ve.

s1d

Here we assume that the plasma is cold, i.e.,Te=0 andme isa phenomenological scalar viscous diffusion term. The effec-tive viscosity term used is very qualitative and we use vis-cosity as an effective energy sink. The continuity equationmay be written as

]n

]t+ = · snved = 0. s2d

In addition, we need the Maxwell equations

= 3 E = −1

c

]B

]t, s3d

= 3 B =4pJ

c+

1

c

]E

]t. s4d

First, let us look for linear waves both in homogeneousand inhomogeneous systems. For this we will use the linear-ized equations above and denote the perturbed and equilib-rium variables by the subscripts 1 and 0, respectively. Ac-cordingly, we can write the momentum, continuity andMaxwell’s equations as

−m

e

]v1

]t= E1 +

1

cv1 3 B0, s5d

]n1

]t+ v1 · = n0 + n0 = ·v1 = 0, s6d

v1 = −c

4pn0e= 3 B1 +

1

4pn0e

]E1

]t. s7d

To demonstrate the waves in homogeneous plasma we taken0=n00, a constant. Then, after some algebra we can reducethese equations into equations with two variables, namely,fluctuating magnetic field and fluctuating density. In theusual method, assuming the solutions of the perturbations tobe of the formB1, n1,expf−isvt−k ·r dg, where v is thefrequency andk is the wave vector, we can find the coupledequations for density and magnetic field perturbations as

Sc2k2 − v2

vp2 + 1DB1

B0=

n1

n0, s8d

svp2 + vc

2 − v2dn1

n0= vc

2B1

B0. s9d

Note here that, Eq.(8) corresponds to an electromagneticwave in absence of density fluctuations whereas Eq.(9) rep-resents the upper-hybrid oscillation in the absence of mag-netic fluctuations. In general, the linear dispersion relation inthe homogeneous electron plasma can be expressed as

Sc2k2 − v2

vp2 + 1D =

vc2

vp2 + vc

2 − v2

. s10d

In the limit v2!k2c2, svp2+vc

2d we get

k2c2

vp2 +

vp2

vp2 + vc

2 =v2vc

2

svp2 + vc

2d2 .

The above relation can be further simplified forvp,vc andthe simplified dispersion relation looks like

v2 = vp2 + k2VAe

2 , s11d

which indicates that the simple plasma oscillation is modi-fied by the electromagnetic effect.

Next, let us consider the inhomogeneous plasma. In thiscase, the equilibrium density distribution is assumed asn0sxd /n00,s1−x/Lnd−1, wheren00 is constant andLn is thescale of density variation withx/Ln!1. Following the sameprocedure outlined for the homogeneous case, we find thelinear dispersion relation as

FSk2c2 − v2

vp2 DS1 +

ikx

k2LnD + S1 −

v*

v−

vv*

k2c2 DG3Fvp

2

vc2 + S1 −

v2

vc2DS1 +

ikx

Lnk2DG

= S1 −v

vc

ky

Lnk2 +

ikx

Lnk2DF1 −

v*

vS1 −

v2

vc2DS1 −

v2

k2c2DG .

s12d

Here we also used the other Maxwell’s equations, viz.,= ·E1=−4pen1 and=3E1=−c−1]B1/]t. Once again we re-state that the definition of various parameters used in theabove equation is as follows; the plasma frequencyvp

=s4pn00e2/md1/2, the electron cyclotron frequencyvc

=eB0/mc, and the drift frequencyv* = kydeVAe/Ln. The elec-tron skin depthde=c/vp will be taken as the unit of the

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spatial scale in this problem andVAe=B0/ s4pmn00d1/2 de-notes the electron Alfven velocity.

For homogeneous plasma, i.e.,Ln→`, v* →0, and werecover the homogeneous dispersion relation Eq.(10). Wehave seen this dispersion relation can support high frequencywave, namely, electron plasma wave modified by the linearelectron magnetosonic wave.

Next let us look in the low frequency regime wherev,v* ,kc,vp,vc. We are interested for the wave of largeradial scale length, i.e.,kx!1, under such conditions the dis-persion relation[Eq. (12)] reduces as

k2c2

vp2 S1 +

vc2

vp2D <

v*

v− 1. s13d

In the limit vc,vp we get

v <v*

1 + k2dc2 , s14d

which is the dispersion relation of electron magnetic driftwave.

In this work we are interested in this low frequency drift-like mode and its nonlinear developments. To derive the non-linear evolution equation in the low frequency regime weneglect the displacement current. Whether it is justified toignore the displacement current or not we shall see later.From Eqs.(2) and (4) we can say that the density perturba-tion may be ignored, which is the EMHD approximationcompatible with= ·J=0 condition.

Keeping this in mind and taking curl of Eq.(3), usingMaxwell’s equations we obtain a self-consistent nonlinearequation for the evolution of electron vorticity.

S ]

]t+ ve · = DsV − Vcd + f= ·vegsV − Vcd

− fsV − Vcd · = gve = me¹2V, s15d

where V= = 3ve is the electron vorticity,Vc=eB /mec isthe cyclotron vorticity. The electron velocity can be obtainedfrom Eq. (4) as

ve = −c

4pen0sxd= 3 B. s16d

Since EMHD is a theory to describe the motion of magne-tized electrons in the presence of self-generated as well asexternally applied magnetic field, in this problem we expressthe total magnetic field as

B = ezfB0 + bsx,y,tdg. s17d

Here B0 is the average uniform magnetic field andb is thefluctuation. Note that the magnetic field is entirely in theaxial ez direction. To be sure, this model ignores the mag-netic field perpendicular to the axial direction for simplicity.With the magnetic field given in Eq.(17) we find =3B=−ez3 = b. When the variations are confined in thex-yplane only, then in the vorticity Eq.(15) the third term is zeroand the leading order equation may be expressed as

S ]

]t+

c

4pen00ez 3 = b · = DF mec

2

4pen00e2¹2b − sB0 + bdG

+c

4pen00

1

Ln

]b

]yF mec

2

4pen00e2¹2b − sB0 + bdG

< me¹2F mec

2

4pn00e2¹2bG , s18d

where we have taken the same equilibrium density distribu-tion as used in the linear theory. In the velocity and vorticityexpression we have assumedc/4pne<c/4pn00e in the lead-ing order. Also note that the second term in the evolution ofthe magnetic field arises due to the nonzero divergence ofelectron velocity. To rewrite the Eq.(18) in dimensionlessvariables the normalization used is as follows: The magnetic

field is normalized byb;sb/B0dsLn/ded. The coordinatevariablesx and y are normalized byde, the electron skindepth. The time variable is normalized byLn/VAe. The nor-malized viscosity is represented asm=meLn/ sVAede

2d.The equations forbsx,y,td in the normalized variables

can then be written as

S ]

]t+ ez 3 ='b · ='Dsb − ¹'

2 bd +]b

]y+

de

Lnb

]b

]y

= − m¹'2 ¹'

2 b, s19d

where ='; ex] /]x+ ey] /]y. The above equation is simpli-fied assumingk'

2 !1 in the last term in the left-hand side.Equation(19) is the basic equation we are going to study inthis paper. Note here thatb acts like a stream function de-scribing the poloidal electron flowez3='b, which is pro-portional to the poloidal current density.

Here let us look for the linear wave from Eq.(19) inabsence of dissipations. If we switch off all the nonlinearterms in Eq.(19) we may get the reduced linear equation as

]

]tsb − ¹'

2 bd +]b

]y= 0. s20d

Now assuming the solution of magnetic field perturbations to

be of the formb,expf−isvt−k ·r dg we can find the localdispersion relation for the electron magnetic drift wave innormalized variables as

v =ky

s1 + k'2 d

. s21d

We can write it in unnormalized variables as

v =kydeVAe

Ln

1

1 + k'2 de

2 ;v*

1 + k'2 de

2 . s22d

This is the exactly same dispersion relation that we obtainedin the linear calculations. Note that to obtain Eq.(19) wehave neglected the displacement current in Maxwell’s equa-tions, whereas in the linear dispersion relation we kept dis-placement current. It should be mentioned that in the lowfrequency regime the result is not affected by including orignoring this term. Therefore, if we are not interested in thehigh frequency regime, ignoring the displacement current

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does not alter the essential physics of the mode.This dispersion relation displayed as Eq.(22) resembles

that of electrostatic drift wave. To compare this result wemay recall the electrostatic drift wave dispersion relation as

v =kyrscs

Ln

1

1 + k'2 rs

2 , s23d

wherers=cs/Vci andcs=ÎTe/mi. Note here that, in the right-hand side of Eq.(23), if we replace the ion mass by electronmass and kinetic pressures,nTed by magnetic pressures,B0

2d then we get back Eq.(22). This is equivalent to sayingthat in the electrostatic case the wave dispersion scale Lar-mor radius is replaced by the electron skin depth and thesound speed is replaced by the electron Alfven velocity.Since only electrons are taking part in the dynamics and thewave field is the magnetic fluctuations we call this wave the“magnetic electron drift wave.” Like the electrostatic driftwave this wave also propagates perpendicular to the direc-tion of inhomogeneity as well as the magnetic field.

III. CONSERVATION LAWS AND VORTEX LIKESOLUTIONS

In the former section, Eq.(19) is inherently a nonlinearequation and the magnetic turbulence in this regime may bestudied by this model. To consider the turbulence by thismodel it is convenient to study the properties of Eq.(19).Since it has close similarities to the Hasegawa–Mima equa-tion, we can easily find the invariants, which are energy andmean-square vorticity. The energy integral of Eq.(19) ininviscid case may be written in terms of various dimension-less variables as

E =1

2E fb2 + s='bd2gdV,

and the enstrophy integral

K =1

2E fs='bd2 + s¹'

2 bd2gdV.

In the energy expressionb2 is the magnetic energy ands='bd2 is the electron kinetic energy andK is the sum of thekinetic energy and the squared vorticity. These invariantsmight help us to determine the energy cascade direction inEMHD turbulence.11

Next, we study the coherent vortexlike structures due tothe nonlinearly saturated state of the magnetic electron driftmode. First, we look for a two-dimensional vortex solutionneglecting the scalar nonlinearityb]b/]y and viscous dissi-pation term. We can neglect the scalar nonlinearity whenkyLn!1. With this assumptions Eq.(19) becomes

S ]

]t+ ez 3 ='b · ='Dsb − ¹'

2 bd +]b

]y= 0. s24d

The traveling wave solutions are found by definingj=y−ut, whereu is the perpendicular speed of the vortex, andlooking for a solution of the form

bsx,y,td = bsx,jd.

Substituting] /]t;−u] /]j in Eq. (24), we obtain the equa-tion in terms of Poisson bracket notation as

fb − ux,b − ¹'2 b − xg = 0, s25d

wherefp,qg;]xp]jq−]jp]xq. The general solution may bewritten as

b − ¹'2b − x = Fsb − uxd, s26d

whereF is an arbitrary function of its argument. This is theequation describing the nonlinear magnetic electron driftvortex. We look for a localized solution of this equation andtherefore proceed exactly like standard electrostatic driftwave.12 The solution is obtained insr ,ud coordinates. Thechoice ofF for r .a is made by noting thatb, ¹2b→0 asj→` for all x so thatFs−uxd;x/u. For r ,a, F is chosenarbitrarily to ensure¹2b to match properly atr =a. Solvingseparately in regionsr .a and r ,a, we get

bsr,ud = HC1K1sbr/adcosu if r . a

C2r/a cosu + C3J1sgr/adcosu if ra,J

where r2=x2+j2, u=tan−1sj /xd, C1=ua/K1sbd, C2=uas1+b2/g2d, C3=−uasb2/g2d /J1sgd. Another unknown param-eter g is determined by the nonlinear dispersion relationgJ1sgdK2sbd+bJ2sgdK1sbd=0 andb is given byb2/a2=s1−1/ud. HereJ1 is the first-order Bessel function of the firstkind andK1 is a first-order modified Bessel function of thesecond kind. The solutionsbsr ,ud given above is known asthe dipole vortex.

Note thatz3='b·=¹'2 b is responsible for the satura-

tion of the wave—which physically means that the convec-tion of the electron vorticity nonlinearly saturates the mag-netic electron drift wave. In absence of nonlinear termz3='b·=¹'

2 b in Eq. (24), the arbitrariness of the functionalform F will not arise and therefore we lose the freedom tochoose the functional form ofF in two different regions,namely,r .a and r ,a. We thus have only a single regionwith ¹'

2 b=b2b, which does not give well behaved and con-fined vortex solutions. The two region confined solutions ofthe type discussed here were first written by Larichev andReznik13 for the Rossby wave equation and describe coher-ent dipole vortices as nonlinear stationary state of the equa-tion. In this dipole vortex solution the parametersa, u maybe chosen independently by satisfyingb2/a2.0.

Next we can discuss the axisymmetric vortex solution ofEq. (19) without dissipation. If we look for the exactly axi-symmetric solution in the frame moving with the vortexstructure thenbsx,y,td=bsrd, where r =Îsx2+j2d and j=y−ut. In this case the vector nonlinearityfb,¹'

2 bg vanishes.Writing ¹'

2 b in axisymmetric variables, Eq.(19) may bewritten as

1

r

d

drSr

db

drD = S1 −

1

uDb − S de

2uLnDb2. s27d

Note that in the right-hand side of Eq.(27) if the b2 term isabsent then from the knowledge of the dipole solution ob-tained before we can say that the solution is a decaying

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Bessel functionfb,e−lr /Îr ,l=Îs1−1/udg. Now if the sec-ond term containingb2 is present then this decaying solutionis valid whenubu!2su−1dde/Ln. With the presence of theb2

term if b.2su−1dde/Ln then the sign of the right-hand sidemay change from positive to negative and a physically ac-ceptable solution can be found. To solve Eq.(27) one can usean analytical approach based on variational principle

dE LSb,db

drD = 0,

where

L =r

2Sdb

drD2

−r

2S1 +

1

uDb2 +

r

3S de

2uLnDb3.

This method was first used by Petviashvili and Pokhotelov.14

They took a trial solutionb=bmsechpslr /pd and minimizedthe variational integral with respect to the parameterbm andp. Using this ansatz here, we find

b <4.8su − 1dLn

deFsechH3

4S1 −

1

uD1/2

rJG4/3

. s28d

The above solution is a solitonlike solution with large ampli-tude, sincede is small.

IV. SUMMARY AND CONCLUSION

We have studied the effect of the density gradient onelectron plasma in a stationary ion background. Due to thecompressibility condition of the electron flow velocity, a newmode arises in EMHD plasma which has a similarity withthe electrostatic drift wave. Due to similarity with the lineardispersion relation of the drift wave, we call this the mag-netic electron drift wave. This wave also propagates perpen-dicular to both the density gradient as well as magnetic field.Next we note that the nonlinear evolution of the axial mag-netic fluctuations also looks similar to the electrostatic fluc-tuation equation(Hasegawa–Mima equation). We have stud-ied a coherent nonlinear vortexlike solution of this equation.Axisymmetric monopole vortex as well as short scale dipolevortex solutions are discussed in two different limits. In pres-

ence of both, the nonlinearity analytical solution is difficult.However, we speculate that there might be interaction of themonopole and dipole solutions which can be studied by com-puter simulations. Perhaps, the most important conclusion inthis work is that the “magnetic electron mode” is identifiedand its nonlinear evolution equation is written down here.We have shown that both electrostatic and magnetic fluctua-tions can be described by a nonlinear partial differentialequation like the Hasegawa–Mima equation. Therefore, apartfrom the physical scaling in time and space, these two wavesmust have similar properties in both the linear and nonlinearregime. We have shown that stationary state vortices may bethe natural choice for the nonlinear solutions. However, nointeractions among vortices are studied. The energy cascadeprocesses also are not studied in the present paper. Theseissues are interesting and we hope to report on them in thefuture.

ACKNOWLEDGMENTS

The authors would like to acknowledge the detailed dis-cussions with Professor P. K. Kaw and the anonymous ref-eree for asking thought provoking questions.

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