Magnetodynamics of a multicomponent (dusty) plasma. II. Magnetic driftwaves in an inhomogeneous mediumLeonid Rudakov and Gurudas Ganguli Citation: Phys. Plasmas 12, 042111 (2005); doi: 10.1063/1.1881513 View online: http://dx.doi.org/10.1063/1.1881513 View Table of Contents: http://pop.aip.org/resource/1/PHPAEN/v12/i4 Published by the American Institute of Physics. Additional information on Phys. PlasmasJournal Homepage: http://pop.aip.org/ Journal Information: http://pop.aip.org/about/about_the_journal Top downloads: http://pop.aip.org/features/most_downloaded Information for Authors: http://pop.aip.org/authors

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Magnetodynamics of a multicomponent „dusty … plasma. II. Magnetic driftwaves in an inhomogeneous medium

Leonid RudakovIcarus Research, Inc., P.O. Box 30780, Bethesda, Maryland 20824-0780

Gurudas GanguliPlasma Physics Division, Naval Research Laboratory, Washington, DC 20375-5346

sReceived 2 November 2004; accepted 1 February 2005; published online 1 April 2005;publisher error corrected 26 April 2005d

The electromagnetic oscillations near the low-frequency cutoffsthe rotation wavesd and theirstability and nonlinear state, analyzed in the companion paperfG. Ganguli and L. Rudakov, Phys.Plasmas12, 042110s2005dg for a homogeneous plasma, are generalized to include inhomogeneityin the equilibrium density and magnetic field. It is shown that in the forbidden frequency band belowthe cutoff the magnetic drift waves appear, which can exist even in a cold plasma. The magnetic driftwave and nonlinear structures associated with it are analyzed and their relevance to astrophysicalplasmas are discussed. It is found that in an inhomogeneous plasma the rotation wave packets cancouple to the magnetic drift waves. The behavior of such structures is governed by nonlinearSchrödinger equation. The spatial scale of the nonlinear structures due to the magnetic drift wavesin astrophysical plasmas such as dense molecular clouds, which are the regions of star creation, isestimated to be around 10 a.u. ©2005 American Institute of Physics. fDOI: 10.1063/1.1881513g

I. INTRODUCTION

In a companion paper,1 henceforth referred to as paper I,we described the origin of a distinctive first-order equilib-rium rotation of the lighter fluids in a multispecies plasma.The characteristic rotation frequency was found to beVr

=sZnH /nedseB/mLcd when the mass density of the heavierspecies dominates. HerenH andne are densities of the heavycomponent and electrons, respectively,Z is the charge stateof the heavy component,B is the magnetic field, andmL isthe light ion mass. We discussed the consequences of thisrotation in a homogeneous plasma and found that the rotationsid introduces a new time scale and a cutoff for waves belowthe rotation frequency,sii d affects waves whose frequency isclose to the rotation frequency,siii d leads to plasma energi-zation and amplification of the ponderomotive force, andsivdleads to the formation of structures which are described bythe nonlinear Schrödinger equation. As a result, the plasmadynamics in a multi-ion plasma around the rotation fre-quency can be significantly affected. In this paper we extendthe analysis to an inhomogeneous system and to frequencyrange smaller than the rotation frequency and show that themagnetic drift waves can exist in the band gap below therotation frequency, where, because of the cutoff phenom-enon, the magnetosonic and Alfvén waves cannot penetrate.Consequently, unlike the situation in a homogeneous plasma,a nonlinear Schrödinger equation can be formed even forstationary dust in an inhomogeneous system due to the cou-pling of the rotation waves with the magnetic drift waves.

Magnetic drift modes2–5 are distinct from the universalelectrostatic drift modes which do not exist in a cold plasma.We will discuss the modification of the magnetic drift modesin a dusty plasma and establish its relationship with the ro-tation modes. The magnetic drift waves is one of the many

electromagnetic waves that have recently been discussed6,7

in the context of dusty plasmas but was first addressed some35 years ago with regard to electron magnetohydrodynamicssMHDd.2–5 For a cold electron-ion plasma equilibrium thatincludes a magnetic fieldB0sxd in thez direction and densitynsxd, it was shown that in the frequency range between elec-tron and ion cyclotron frequencies there is the magnetic driftwave where the magnetic perturbation oscillates with the fre-quency v=kycB0/4penLns1+k2de

2d. Here de=c/vpe is theelectron inertial scale and1/Ln=d lnsrd /dx. For motionlessions these waves are stable in the cold plasma limit. Detaileddiscussion in the context of electron MHD physics was pro-vided in a review paper.5

Magnetic drift waves in a dusty plasma is important be-cause of space and astrophysical relevance. Dust particlesoften exist in space and astrophysical environments8–12 andare distributed over radiusa, asndsad,1/a3−1/a4, wherend

is the dust density. These grains are usually charged and canchange the electrical property of the medium.9,11,12Averagedover particle distribution the mass density of the dust com-ponent mdnd is determined by larger particles while thecharge densityZend depends on smaller ones wheremd is themass of the dust component. The charges are mostly nega-tive. Equilibrium charge of dust grains is estimated in a man-ner similar to a Langmuir probe at floating potential.13 Forastrophysical environment the dust charge state can be esti-mated asZ>−0.3aT−1,9 where a is given in micrometerand the plasma temperatureT in Kelvin. For a,1 mm scor-responding to dust mass,md,4310−12 g, T=10 K, thecharge stateZ=4. On the astrophysicalstemporal and spatialdscale the plasma electrons and ions are magnetized while thedust particles may be unmagnetized because of the dragforce or for those processes which are faster than dust cyclo-tron time.8–12 In the interstellar medium on average most of

PHYSICS OF PLASMAS12, 042111s2005d

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the matter is in the form of cold neutral gas.8–10The net massof the dust component is only a few percent of that of theneutral gas. In a dense molecular cloud the typical neutraldensitynn,103 cm−3, the magnetic fieldB,30 mG, andT,10 K. The ionization in the dense molecular clouds is dueto natural radioactivity and cosmic radiation. Typical ratio ofions to the neutralsni /nn,10−6, the ratio of charged dust toionsZnd/ni ,10−4, andmdnd,104mini. The plasma pressurein weakly ionized dense molecular clouds is negligible com-pared to the magnetic pressure, which is on order of theneutral gas pressure.8–10

II. MAGNETIC DRIFT WAVE DISPERSION RELATION

We investigate the magnetic oscillations in an inhomo-geneous plasma in the frequency range below the ion cyclo-tron frequency in a typical astrophysical environment whoseparameters are briefly discussed in the preceding section. In-stead of the general approach based on the plasma dielectrictensor we simplify the analysis by focusing specifically tothe magnetic drift waves with the assumptions thatZnd!ne,which is equivalent tone<ni, and ndmd@nimi. We willevaluate the Hall electric field from the perpendicular current

and use the Faraday law,¹W 3EW =−c−1]BW /]t, to eliminate theelectric field and obtain the dynamic magnetic field equation,which readily yields the dispersion relation. We will evaluateand analyze the dispersion relation starting from the simpleinertialess and collisionless cases and then progressively addmore physics until we achieve the realistic physical condi-tions relevant to the dense molecular clouds. This will allowus to highlight the physics and make the calculations trans-parent and tractable.

A. Collisionless and inertialess limits

To demonstrate the essential physics of the magneticdrift waves let us first consider the simplest case in which theinertial terms of electrons and ions can be ignored and thedust is considered to be stationary. The equations of motionfor the componentss=i ,ed simplifies to

cEW + VW s 3 BW = 0. s1d

Multiplying Eq. s1d by esns and adding them we get

esni − nedcEW + esnivW i − nevWed 3 BW = eZndcEW + jW 3 BW = 0.

s2d

Equation s2d follows from the quasineutrality condition,−Znd=ne−ni, which couples the ion and electron motion, andthe definition of current,jW=esnivW i −nevWed. Now we calculatethe Hall electric field from Eq.s2d,

EW = −jW 3 BW

cen=

1

8pen„¹W B2 − 2sBW ·¹W dBW … . s3d

We introduce the notationn=Znd to emphasize that theanalysis in this section and Secs. II C, III A, and III B areidentical to the usual Hall plasma analysis ifn;−ne is sub-stituted in the classical Hall plasma formalisms such as thosein Refs. 2–5. Hence, much of the physics developed for the

classical Hall plasmas becomes immediately applicable tothe dusty plasmas.

Curl of Eq. s3d along with the Faraday law leads to thenonlinear equation for the magnetic field,

]BW

]t= ¹W 3

jW 3 BW

en

=c

4peF ¹W n

2n2 3 ¹W B2 + ¹W 3 S1

nsBW ·¹W dBWDG . s4d

Local approximation with magnetic perturbationsB1

,exps−ivt+ ikxx+ ikyy+ ikzzd, yields the dispersion relationfor modified whistler and magnetic drift waves in a coldinhomogeneous plasma. Rewriting the vector equations4d in

components along with the constraint that= ·BW = ikzB1z

+ ikyB1y=0, where we assume for simplicity thatkxB1x

!kyB1y, we get

− ivB1z =c

4pen0FikyB0B1z

dn0

n0dx+ kzkyB0B1xG , s5d

− ivB1x =c

4pen0F− kzkyB0B1z + ikyB1x

dB0

dxG , s6d

where we have neglected terms of the order ofskz/kyd2. Amore general derivation including these terms is given in theAppendix. The subscripts 0 and 1 refer to the equilibriumand fluctuating quantities. Solving this set of equations weget

Sv +kycB0

4peZ2nd02

dZnd0

dxDSv +

kyc

4peZnd0

dB0

dxD = kz

2ky2ld

4Vr02 ,

s7d

where ld=sni0/Znd0dsmic2/4pe2ni0d1/2 and Vr0=sZnd0/ne0d

3seB0/micd are the dust whistler scale1 and rotation fre-quency. Equations7d establishes the existence of normalmodes of oscillation and also that in a cold plasma there isno instability in the local limit. The nonlocal eigenvaluetreatment of a similar problem in the classical electron-ionHall plasma is given in Ref. 2. Forkz=0, Eq.s7d admits twotypes of drift mode oscillations; one of them with frequencyv=vB;kyVB, which are oscillations ofB1x and the otherwith v=vn;kyVn, which are oscillations ofB1z. VB andVn

are defined by

VB = −c

4peZnd0

dB0

dx= −

cB0

4peZnd0

1

LB, s8d

Vn = −cB0

4pesZnd0d2

dsZnd0ddx

= −cB0

4peZnd0

1

Ln. s9d

The gradient scale associated with the magnetic field and thedensity areLB and Ln. The equilibrium gradientB drift VB

originates because of the Hall electric fieldE0x which is sup-ported by thej03B0 force as in Eq.s2d. Thus, if B0 is uni-form thenE0=0. In this paper we consider local limit so thatkxLn,kxLB@1. In the rest of the paper we will refer to thedrift mode branch due to density inhomogeneity, i.e.,v=vn;kyVn, as the magnetic drift waves. Instability of the

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magnetic drift waves in a cold plasma with magnetic fieldinhomogeneity can appear only if the heavy particle motionis included as shown in Sec. II C.

B. Effects of ion inertia and ion-neutral collisions

In this section we generalize Eq.s1d for ions by includ-ing ion inertial term and ion-neutral collision frequencyni,

]vW i

]t+ svW i ·¹W dvW i =

e

mi

sEW + vW i 3 BW /cd − nivW i . s10d

We solve Eq.s10d using an iterative procedure withvW i =cEW

3bW /B as the first approximation. DefiningbW =BW /B we obtain

vW i =cEW 3 bW

B+

1

VibW 3 FS ]

]t+ niDcEW 3 bW

B

+ ScEW 3 bW

B·¹W DcEW 3 bW

BG . s11d

Since the Spitzer ambipolar magnetic field diffusion in denseweakly ionized molecular clouds is limited by the ion-neutralcollision,8,10 we neglect the inertia and collision for the elec-tron fluid to obtain

vWe =cEW 3 bW

B. s12d

From Eqs.s11d and s12d we can obtain the perpendicularcurrent to be

j = esnivW i − nevWed

= eZndcEW 3 bW

B+

eni

VibW 3 FS ]

]t+ niDcEW 3 bW

B

+ ScEW 3 bW

B·¹W DcEW 3 bW

BG . s13d

Linearizing Eq.s13d we obtain

jW1 = eZnd0cEW 1 3 bW

B0+

eni0

VibW 3 Ss− iv + nid

cEW 1 3 bW

B0D .

s14d

For simplicity we have assumed in Eq.s14d that the back-ground magnetic field is uniform so thatVB=0 sequivalently,E0=0d. Equationss13d or s14d is the “Ohm’s law” for a dustyplasma. Solving Eq.s14d we obtain an expression for theperturbed electric field,

EW 1 =c−1

Vr02 − v2

Vr02 jW1 3 BW 0 + ivVr0BW 0 3 jW1 3 bW

eZnd0, s15d

where v=v+ ini and Vr0=sZnd0/ne0dVi is the rotation fre-quency. Taking the curl of Eq.s15d and using the Faradaylaw we obtain the generalization of Eq.s4d to be

]BW 1

]t= ¹W 3

c

Vr02 − v2

3Vr0

2 s¹W 3 BW 1d 3 BW 0 + ivVr0BW 0 3 ss¹W 3 BW 1d 3 bWd4peZnd0

.

s16d

For kz=0, thez component of Eq.s16d in the local approxi-mation sukuLn@1d leads to the dispersion relation

v = kycB0

4pe

d

dxF Vr0

2

Znd0sVr02 − v2dG −

k'2 VA

2v

sVr02 − v2d

, s17d

where VA=B0/Î4pnine0, k'2 =kx

2+ky2. In the homogeneous

collisionless limit the first term on the right-hand side of Eq.s17d vanishes and we recover the dispersion relation for therotation waves forkz=0,1,14

v2 = Vr02 + k'

2 VA2 . s18d

As discussed in paper I and Ref. 15,Vr0 represents a cutofffor waves in the frequency rangeVr .v.Vd. Consequently,the magnetosonic and Alfvén waves cannot propagate in thisband gap. However, in an inhomogeneous plasma the mag-netic drift waves can exist in this band gap. In the frequencyrangeVr @v we can neglectv compared toVr in denomi-nator of right-hand side of Eq.s17d and obtain the dispersionrelation for magnetic drift waves including the effects of ioninertia and ion-neutral collision,

v = vn ;kyVn − inik'

2 ld2

1 + k'2 ld

2 , s19d

where the dust whistler lengthld is defined below Eq.s7d.Comparing Eq.s19d with the inertialess limit discussed inSec. II A, we see that the effect of the ion inertia and ion-neutral collision on the magnetic drift waves is to introduce afactor 1/s1+k'

2 ld2d to the wave phase speed and collisional

dissipation. This is similar to the magnetic drift waves in theclassical electron-ion Hall plasma.5 Also, from the isomor-phism of the formalism with the classical Hall analysis inwhich the heavy particles are motionless, we can concludethat the inclusion of the background magnetic field inhomo-geneity would not lead to an instability and affect the disper-sion relation Eq.s19d in any significant way.5,7 However, inthe presence of the heavy particlesdustd motion the magneticfield inhomogeneity is important and can lead to the mag-netic drift wave instability as discussed in Sec. II C below. Amore general dispersion relation for arbitrary wave fre-quency and wave vector orientation is given in the Appendix.

C. Effects of dust inertia

We now take into account the dust motion and add themomentum balance equations of all the components to ob-tain

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minidvW i

dt+ mdnd

dvWd

dt= c−1JW 3 BW + FW , s20d

where the net currentJ is the sum of the current due to theHall electric field j as defined in Eq.s2d and the dust currentand is given by

JW =c

4p¹W 3 BW = jW − Zendvd = esnivW i − nevWe − ZndvWdd. s21d

FW =−ndmdndvWd<−kvntmnnna2lndvWd where “k¯l” implies av-

erage over the grain sizes is the drag force acting on dust inthe environments such as the dense molecular cloudssmn, nn,vnt are the mass, density, and thermal velocity of the neutralmolecules, respectivelyd.8–12 Gravity, being a long-rangeforce, can be ignored for fast and small-scale dusty plasmamotion. For simplicity the ion-neutral collisions are also ne-glected.

We assume that the dust mass density dominates, i.e.,ndmd@nimi. Then linearized equation of momentum balancefi.e., Eq.s20dg and dust continuity are

− ivmdnd0vWd1 = −¹W B0Bz1

4p− ndmdnd0vWd1, s22d

− ivnd1 + ¹W ·nd0vWd1 = 0. s23d

We consider the case wherekz=0 andvd0=0. The contribu-tions from the ion and electron components enter through thecurrent, Eq.s21d, and the quasineutrality condition. Underthese conditions, the Hall electric fieldfEq. s3dg is modifiedto

EW = −sJW + eZndvWdd 3 BW

ceZnd. s24d

Curl of Eq. s24d yields the equation for the magnetic fieldperturbation to be

− ivBW 1 = ¹W 3 F sJW1 + eZnd0vWd1d 3 BW 0 + JW0 3 BW 1

eZnd0

−JW0 3 BW 0

eZnd02 nd1G . s25d

In the local approximation thez component of Eq.s25d leadsto

vB1z = − kycB0

4pesZnd0d2

dsZnd0ddx

B1z − B0Sis¹W ·vWd1d

− kynd1

nd0

c

4peZnd0

dB0

dxD . s26d

Equationss22d, s23d, ands26d yield the dispersion relation

k'2 VAd

2

vsv + indd=

v − kyVn

v − kyVB, s27d

whereVAd=B0/Î4pmdnd0 andVn, VB are defined in Eqs.s8dands9d. Equations27d indicates that the dust motion couplesthe two drift modes with the dust magnetosonic mode. Forcomparison of Eq.s27d with Eq. s19d in Sec. II B, we must

consider the limitukuld!1 andni =0 because in deriving Eq.s27d the ion-inertia and ion-neutral collisions were ignored.If Ln,LB! ldsmdnd0/mini0d1/2 sor equivalently,Vn, VB@VAddthenv can be ignored in the right-hand side of Eq.s27d andits solutions are

v =1

2F− ind ±Î− nd

2 + 4skyVAdd2SVB

VnDG . s28d

If the gradients in magnetic field and dust charge densityhave opposite signs, i.e.,VnVB,0, then the magnetic driftwaves are unstable. In a classical electron-ion Hall plasmathis instability is discussed in detail in Refs. 4 and 5. Thus,we see that the inclusion of the dust motion along with aninhomogeneous magnetic field is essential for an instabilityof the magnetic drift waves in the local limit, which arestable otherwise as shown in Secs. II A and II B.

The instability can spontaneously arise in dense molecu-lar clouds and could play an important role in its dynamics.The macroscopic equilibrium in the dense molecular cloudsis determined by the balance ofj03B0 force, the pressuregradient, and gravityfi.e., c−1j03B0− = snnTnd+G=0, sincethe plasma pressure is negligible compared to the neutralpressureg. Since c−1j03B0= ¹B0

2/8p, magnetic field inho-mogeneity is naturally generated and in a dusty plasma themagnetic inhomogeneity gives rise to aVB, which could begreater than the electron velocity in a classical Hall plasma,vey=c¹xB0/4pene0, by a factor ofne0/Znd0=ni0/Znd0@1.The free energy available in the magnetic field inhomogene-ity and the drifting ions can spontaneously give rise to theinstability described in Eqs.s27d and s28d.

III. NONLINEAR STRUCTURES DUE TO MAGNETICDRIFT WAVES IN DUSTY PLASMAS

In Sec. II we described magnetic drift waves and theirstability in an inhomogeneous dusty plasma. Now we discussthe nonlinear structures that these waves give rise to underdifferent conditions. For simplicity, let us consider a one-dimensional magnetic field,Bzsx,y,td, in an inhomogeneousdusty plasma,Znd=Zndsxd, with stationary dust.

A. Highly collisional plasma „nišv…

We first examine the dynamics of a magnetic drift wavepulse in a highly collisional medium where the ion-neutralcollisions dominate the ion inertia while the dust is consid-ered to be motionless. Hence, neglecting ion inertia in Eq.s13d we get

jW = eZndcEW 3 bW

B+ eni

ni

Vi

cEW

B. s29d

Solving this equation we find the Hall electric field

EW =1

ce

Znds jW 3 BW d + snini/Vid jWB

sZndd2 + snini/Vid2 . s30d

Curl of Eq. s30d leads to the nonlinear equation

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]Bz

]t−

c

8pe¹W B2 3 ¹W

Znd

sZndd2 + snini/Vid2 = ¹W · hi¹W Bz,

s31d

where hi =snimic2/4pe2nid / fsZnd/nid2+sni /Vid2g is the

Spitzer diffusion coefficient corrected for the presence ofdust. Equations31d has been previously studied9,11 in one-dimensional case and a reduction of the Spitzer ambipolarmagnetic diffusion in dusty plasma by the factorsndZ/nid2/ sni /Vid2 has been discussed. However, the physicsrelated to the second term in the left-hand side of Eq.s31dwas not considered. In the limitsndZ/nid2/ sni /Vid2@1, Eqs31d reduces to

]Bz

]t−

Vn

2B0

dBz2

dy= hi

d2Bz

dy2 , s32d

wherehi =nismic2ni /4pe2Z2nd

2d andVn is defined by Eq.s9d.Note that since the dust is stationary,nd=nd0. This is theBurgers equation and it has analytical solution. Magneticfield evolution by the Burgers equation is extensively dis-cussed in the literature, e.g., Ref. 16. Linearizing Eq.s32d wecan obtain the magnetic drift wave dispersion relation, Eq.s19d, without the ion inertial termsi.e., v=kyVn− inik'

2 ld2d. A

magnetic drift wave pulse traveling along a constant densitysurface in a dusty plasma can steepen because of nonlinearityin a manner similar to shock formation in an usual electron-ion MHD plasma. As a result, the pulse can convert into ashock wave,B=Bzsy−Utd, with the front widthhi /U and theshock velocityU,Vn.

3–5,16 This enables the magnetic fieldto penetrate deep into the plasma over distances muchgreater than the skin depth than it would due to Spitzer am-bipolar diffusion.

It is remarkable that the addition of a small dust compo-nent can enable a magnetic shock, described above, to pen-etrate in the plasma faster than the usual compressionalMHD shock if Vn is larger than the dust Alfvén speedVAd

=B/ s4pndmdd1/2.3,4 Also, for Vn@VAd, the dust motion isnegligible. This occurs if Ln!c/vpd, where c/vpd

=smdc2/4pe2Z2nd0d1/2= ldsmdnd0/mini0d1/2. In dense molecu-

lar clouds at 10 K anda,0.1 mm only one electron can beattached to a grain, i.e.,Z,1. Using Refs. 9 and 10 weassume fornn=103 cm−3, B=30 mG ni =10−3 cm−3, Znd

=5.0310−8 cm−3, ni /Vi =10−5. For these dense molecularcloud parameters we estimateld,1 a.u. and c/vpd

,100 a.u. This implies thatVn is larger thanVAd as long asLn,100 a.u. Also, we havesndZ/nid2/ sni /Vid2@1, whichensures that Eq.s32d remains valid. Thus, nonlinearly themagnetic drift waves can significantly enhance the transportof fields in a dusty astrophysical plasma and influence theirdynamics.

To better understand the physics of the magnetic fieldpenetration in inhomogeneous dusty plasma in the form of ashock, it should be remarked that in the frequency rangev@Vi a similar phenomenon occurs in the usual two-component plasma.3–5,16Magnetized particles drifting in Hallelectric field transport the frozen-in magnetic flux and themagnetic energy. They preserve the “frozen-in” conditionsB/n=const in the frame moving with magnetized fluidsd by

drifting along the wave front towards the increasing densityin the magnetic field, which increases in time. However,there is no local frozen-in law in such a wave, in contrast toa conventional MHD where the electron fluid displacementalong the shock front is neglected. In Ref. 17 it was shownthat the phenomenon of shock such as magnetic field pen-etration may exist even in a collisionless homogeneous Hallplasma.

B. Low-frequency collisionless limit „ni™v™Vr…

We now investigate the formation of nonlinear structuresdue to the magnetic drift waves in the limit of collisionlessions and motionless dust. We evaluate the Hall electric fielditeratively. First we set the ion inertial drift current in right-hand side of Eq.s13d to zero. This yields an electric field inthe first approximation which is identical to that described inEq. s3d. Substituting this electric field in the ion inertial driftcurrent in the right-hand side of Eq.s13d we obtain a cor-rected electric field

EW = −jW 3 BW

ceZnd0+

4pld2

c2

] jW

]t+ Vr

s4pd2ld4

c3BZnd0s jW ·¹W d

jW

Znd0.

s33d

Then taking curl of Eq.s33d we obtain, in the approximation=ln B@ ¹ ln Znd0, a nonlinear equation,

]

]tsBz − ld

2¹2Bzd + Vn]Bz

]y+

Vr

Bld4F ]Bz

]y

]

]x−

]Bz

]x

]

]yG¹2Bz = 0.

s34d

This nonlinear equation is similar to the Charney–Hasegawa–Mima equation18 and describes the magnetic driftwaves and localized nonlinear structures due to them. Bylinearizing Eq.s34d, we see that last term is of higher orderand the dispersion relation for the magnetic drift waves, Eq.s19d with ni =0, is recoveredfi.e., v=kyVn/ s1+k'

2 ld2dg.

Vortex-type solutions for Eq.s34d have been extensivelydiscussed.5,6,19 In homogeneous plasma, Eq.s34d describesstationary magnetic filamentsselectrical current vortexesd.The Poisson bracketfs]Bz/]yds] /]xd−s]Bz/]xds] /]ydg iszero for any function ofBz. We choose

ld2¹2Bz = Bz + csBzd, s35d

wherecsBzd is an arbitrary function, which gives boundaryconditionB=Bmax at the center of the filament andBz→B0 atinfinity. Such filaments drift with velocityVnsB0d if theplasma is inhomogeneous.6

C. Resonant frequency limit „vÉVr…

The nonlinear dynamics near the rotation resonanceleads to the nonlinear Schrödinger equation for the system ina manner similar to the homogeneous plasma.1,15 However,there are important differences. A crucial requirement in theformation of the nonlinear Schrödinger equation in a homo-geneous plasma is the inclusion of the slow dust motion,without which it is not possible to achieve a nonlinear fre-quency shift.1,15 The fast time scale is determined by the

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light fluid rotation and the dust magnetosonic oscillationsprovide the slow time. They are coupled by the ponderomo-tive force which is large near the rotation resonance. In aninhomogeneous plasma, however, a nonlinear frequency shiftis possible without the dust motion. In this case, instead ofthe dust magnetosonic waves, the magnetic drift waves canprovide the slow time scale.

Near the rotation resonance in a collisionless inhomoge-neous plasma the term in Eq.s17d responsible for the mag-netic drift waves is negligible because it is proportional to1/Ln compared to the remaining term which is proportionalto the wave vectork andkLn@1. As a result we recover thedispersion relation for the rotation waves in a quasihomoge-neous plasma given by Eq.s18d. Thus, as in paper I, orfollowing the Zakharov20 prescription, i.e., usingv=Vr0

−dV+ i ] /]t in Eq. s18d fv=Vr0+skx2+ky

2dVA2 /2Vr0g, we can

obtain an equation for ion fluid velocity amplitudev1sx

=−iv1sy;v1s,

i]v1s

]t= dVv1s −

VA2

2Vr0S ]2

]x2 +]2

]y2Dv1s, s36d

wheredV is the nonlinear frequency shift and the subscriptsdenotes the slow variation as described in paper I. We nowneed to evaluate the nonlinear frequency shift due to themagnetic drift waves and this is where the physics is differ-ent from that of the homogeneous plasma case. In general wemay express

dV = dSZeBndmicne

D = Vr0SdB

B0−

dne

ne0+

dnd

nd0D . s37d

In a homogeneous plasma, the frozen-in law results in therelation dB/B0=dne/ne0. sNote d represents a second-orderquantity generated due the action of ponderomotive forceresulting from the fast oscillations.d Consequently, the non-linear frequency shift can exist only if the dust motion istaken into account. But in an inhomogeneous plasma,dVmay develop due to nonlinear coupling with magnetic driftwaves which propagates in they direction. Thus, we use] /]x!] /]y in Eq. s36d.

To obtaindE, dBz we need the ion equation of motionalong with the continuity equation and the Faraday law. Therelevant ion equation of motionfi.e., Eq.s13d of paper Ig canbe expressed as

miniS ]vW i

]t+ svW i ·¹W dvW iD = eZndEW + c−1jW 3 BW

= − eZndc−1svW i 3 BW d + c−1jW 3 BW ,

s38d

where the ion velocity given byvW i =cEW 3bW /B. This equationcan also be derived by taking the vector product of Eq.s13dwith BW . When averaged over time much longer than 1/Vr

Eq. s38d leads to

eZnd0dEW − ¹WdBzB0

4p− ¹W

uB1su2

8p− mini0¹W

uv1su2

2= 0, s39d

wherevW1s/c=EW 1s3BW 0/B02 and the last two terms are due to

the ponderomotive force. Equations39d is valid for quasi-stationary solutions for which the ponderomotive force islargely determined by the variation of amplitude of the rota-tion wave packetssee Appendix A of Ref. 1d. Solving fordE

from Eq. s39d we can obtaindvW =dEW 3BW 0/cB02. Using dv in

the continuity equation we obtain an equation fordne,

]dne

]t= −

c

8peF d

dxS 2ne0

Znd0D ]dBz

]y+

d

dxS ne0

Znd0D ] uB1su2

B0 ] y

+d

dxS4pmine0ni0

Znd0D ] uv1su2

B0 ] yG . s40d

Similarly, usingdE in the Faraday law, −dBW =c= 3dEW , weobtain an equation fordBz,

]dBz

]t= −

cB0

8peF d

dxS 2

Znd0D ]dBz

]y+

d

dxS 1

Znd0D ] uB1su2

B0 ] y

+d

dxS4pmini0

Znd0D ] uv1su2

B0 ] yG . s41d

Now we can calculate the nonlinear frequency shift Eq.s37dfor the wave packet moving with velocityU and dBz, dne

, fsy−Utd so that] /]t;−U] /]y. Using these, the nonlinearfrequency shift can be expressed as

dV = Vr0SdBz

B0−

dne

ne0D

= − Vr0cB0

4peZnd0U

dne0

ne0dxFdBz

B0+

uBlsu2

2B02 +

ni0uv1su2

2ne0VA2 G ,

s42d

dBz

B0=

cB0

8pesU − VndF d

dxS 1

Znd0D uB1su2

B02

+d

dxS nio

Znd0D uv1su2

ne0VA2G . s43d

Equationss36d, s42d, and s43d form a complete set that de-scribes the behavior of the self-organized wave packet filledby the rotation waves. This shows that the coupling of thefast rotation waves with the slow magnetic drift waves canlead to the nonlinear Schrödinger equation forkz=0 even forstationary dust in a cold inhomogeneous plasma.

IV. CONCLUSION

In this paper we generalized our work reported in paperI and in Ref. 15 to an inhomogeneous plasma and combinedit with ideas developed earlier.6 We included spatial variationin the plasma as well as ion-neutral collisions which are typi-cal for astrophysical dusty magnetoplasma. We studied themagnetic drift waves in a dusty plasma and found that thedust inertia is essential for these waves to be unstable. We

042111-6 L. Rudakov and G. Ganguli Phys. Plasmas 12, 042111 ~2005!

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then, investigated the nonlinear consequences of the mag-netic drift waves to dusty plasmas under various conditions.

We found that in a collisional dusty plasma a magneticdrift wave can steepen to form a shock, which can transportthe magnetic field into the plasma at a rate much faster thanthe magnetic diffusion rate. The nonlinear analysis in thisregime led to another important result that the local frozen-incondition could not be fulfilled as it was in the context of theclassical electron-ion Hall physics.3–5 This breakdown of theMHD frozen-in law on the scale,10 a.u. can affect cosmicturbulence spectrum, magnetic field reconnection,21 gravita-tional collapse phenomena,9 and astrophysical dynamoefficiency.22

We found vortex-type structures of the dust whistlerscale sizeld which are described by a nonlinear Charney–Hasagawa–Mima-type equation for the magnetic drift waves.It was also found that rotation wave packet can couple to themagnetic drift waves. Behavior of such structures is gov-erned by the nonlinear Schrödinger equation. The nonlinearfrequency shift in an inhomogeneous plasma can be achievedby the magnetic drift waves.

We have described these phenomena using a simple dustmodel where the dust particles are considered to be of thesame size and with the same magnitude of charge. In realitythe dust component has a size distribution. The mass densityof dust is defined by the particles of the largest size while thecharge is determined by the smaller size particles. Thus theproblem should be considered by introducing a distributionfunction of dust particles over size. We neglected the thermalpressure of the plasma components since in weakly ionizeddense molecular clouds they are negligible compared to theneutral gas pressure, which is large and balances gravity andmagnetic forces.

We discussed the instability of magnetic drift waves.However, there are other important dusty plasma instabilitiesof relevance to astrophysics that need attention. For example,collapse of dense molecular clouds with dust and magneticfield under gravity can create protostellar disk whose rotationaccelerates in order to conserve the angular momentum. Thedifferential rotation of the disk with initial axial magneticfield could be a potential source for the instability which canlead to the formation of spiral magnetic field in a processknown as kinematic dynamo. It is an important issue in thestar creation process. The magnetorotational instability of theastrophysical plasma was analyzed by taking into accountthe classical Hall effects23 and recently Rudakov applieddusty Hall physics to rotating cylinder with helical magneticfield sBw Bzd and demonstrated that the presence ofBw playsa stabilizing role.22

Experimental investigation of the cutoff and rotationresonance phenomena and magnetic drift waves in a labora-tory dusty plasma is practically impossible in devices cur-rently in use in most of the laboratories involved in suchbasic experiments. For the dusty plasma parameters as thecharge and mass of a dust particlesnd,106/cc, Z,104,md,10−12 g a hydrogen plasma density 1013/cc, the dustinertial scalec/vpd is ,30 km and the dust whistler lengthldis ,100 m. These dimensions are prohibitive and make theexperiment impractical. However, it may be possible to de-

sign experiments if instead of dust-ion-electron plasma amixture of light and heavy gases, e.g., singly ionized hydro-gens80%d and xenons20%d, is used. Then the mass densitywill be primarily determined by xenon and the charge den-sity by hydrogen. In such a plasma with density 1013/cc theheavy particle inertial scale will be,300 cm and the heavywhistler length is,30 cm. The magnetic field,300 G issufficient to magnetize the electrons and light ions. Thesedimensions make the experimental investigation feasible.

ACKNOWLEDGMENTS

This work was supported by the National Aeronauticsand Space Administration and the Office of Naval Research.

APPENDIX: ALFVÉN AND MAGNETOSONIC WAVESIN A DUSTY INHOMOGENEOUS PLASMA

Recently, the linear and nonlinear properties of electro-magnetic waves in the presence of dust were discussed.24–26

Simplifying assumptions employed in these papers resultedin the loss of the rotationscutoffd frequency and the physicsassociated with it that we emphasized.1,15 Specifically, theloss of the rotation frequency in Refs. 24–26 was due to animplicit assumption of high frequency and short wavelengthlimit in the analysis. We demonstrate this by following thetechnique used in Ref. 26 to obtain a more general dispersionrelation with arbitrary propagation angle in a collisionlessplasma with stationary dust grainssi.e., v@Vdd than that isprovided in Sec. II. As shown in Sec. II, the electric field ina dusty plasma may be represented in terms of current. If thebackground magnetic fieldB0z is homogeneous butZnd0 is afunction of x and the plasma is collisionless then Eq.s15dcan be expressed as

EW 1 = −c−1B0

sVr0z − v2deZnd0

ScVr02

4ps− ¹W B1z + ikzBW 1d

+ ivVr0s jW1 − BW 0j1z/B0dD , sA1d

where we assumedVi @v. Curl of Eq. sA1d leads to theequation for vectorB1. Thez component is given by

FsVr02 − v2dS1 −

kyVn

vD + k2VA

2GB1z

=kzVA

2Vr0

vs¹W 3 BW 1dz =

kzVA2Vr0

vs− ikyB1xd, sA2d

wherekx!ky is assumed and

Vn =cB0

4pe

d

dxS Vr0

2

sVr02 − v2dZnd0

D .

Following Ref. 24 we introduce vector potentialA suchthat

E1 = − c−1 ] AW 1/]t − ¹W w1, sA3d

042111-7 Magnetodynamics of a multicomponent ~dusty! plasma. II.... Phys. Plasmas 12, 042111 ~2005!

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BW 1 = ¹W 3 AW 1, ¹W ·AW 1 = 0, jW = −c

4p¹W 2AW 1. sA4d

Because the electron mobility along the magnetic field ishigh, the electric fieldE1z in the low-frequency oscillations isnegligible. FromsA5d we get

w1 = A1zv/kzc. sA5d

Now we calculate= ·E1 using Eq.sA1d to obtain

sVr02 − v2dk2cw1 = Vr0k

2VA2B1z + ikzVnsVr0

2 − v2dB1x

− vkzk2VA

2A1z, sA6d

Since kx!ky, the ¹W ·AW = ikzA1z+ ikyA1y+ ikxA1x=0 conditiongives A1y=−kzA1z/ky. Using this we expressB1x= ikyA1z

− ikzA1y= iA1zk2/ky, wherek2=ky

2+kz2. SubstitutingBlx in Eq.

sA2d we solve forBlz in terms ofA1z. EquationsA5d givesw1

in terms ofA1z. Using these in Eq.sA6d we obtain the dis-persion equation

FsVr02 − v2dS1 −

kz2

ky2

kyVn

vD + kz

2VA2GFsVr0

2 − v2d

3S1 −kyVn

vD + k2VA

2G =Vr0

2

v2 kz2k2VA

4 . sA7d

This is the dispersion relation for electromagnetic waves inthe frequency rangeVi @v@Vd in a collisionless inhomo-geneous plasma. It reduces to the dispersion relation Eqs.s6dand s7d in Ref. 15 in the homogeneous limit, i.e.,Vn=0.Equation sA7d describes both magnetosonic and Alfvénwaves and their coupling in a multispecies plasma. It alsodescribes the rotation waves in an inhomogemneous plasma.The left-hand side of Eq.sA7d is a product of two terms. Thefirst term is due to oscillations ofB1x while the second is dueto B1z. The dispersion relation in Onishchenkoet al.26 can berecovered if the first term is set equal to zero along with thev@Vr0 assumptionfalthough they missed a term propor-tional to Vr

2 in their dispersion relation; see Eq.sA9dg. Thisrequires a decoupling of the two terms in the left-hand sideof Eq. sA7d which can occur only when the right-hand sideof Eq. sA7d is negligible. The right-hand side of Eq.sA7d issmall for high frequency, i.e.,v@Vr0, which also corre-sponds to short wavelengths,kVA/Vr0=kld@1. Hence, themathematical form of the dispersion relation for the shearAlfvén waves in Refs. 25 and 26 implies an implicit assump-tion thatBlx prevails overB1z, which corresponds to the shortwavelength high-frequency limit. This tacit assumption re-sulted in the loss of the cutoff phenomenon in Refs. 25 and26.

Shukla and VarmasSVd sRef. 27d consider the low-frequency limit, i.e.,v!Vr0, andkz=0 and address the elec-trostatic modes in an inhomogeneous dusty plasma and ob-tain the dispersion relationv=VSV=Vrky/Lnskx

2+ky2d. In this

paper and in our previous papers6,7 and also in a recent paperby Shukla28 the low-frequency magnetic drift modes wereconsidered. An important question is whetherVSV representsa new mode or is a limiting case of the magnetic drift mode?The second term in the left-hand side of Eq.sA7d for kz=0andv!Vr0 leads to

v =kyVn

1 + skx2 + ky

2dld2 = VSV skx

2 + ky2dld

2

1 + skx2 + ky

2dld2 . sA8d

We can conclude from Eq.sA8d that the so-called Shukla–Varma frequency is the short wave limit of the well-knownmagnetic drift frequency in an inhomogeneous plasma asgiven in Eq. s19d. This point was missed by Shukla andVarma because of their assumption that the electric field isentirely of electrostatic origin. In an inhomogeneous plasmathe Hall electric field includes a part that is of electromag-netic origin. This can be easily inferred from the curl of theelectric field from Eq.sA1d for kz=0 andv!Vr0. Clearly,the electric field is not purely electrostatic, because there is a

nonzero inductive part given by¹W 3EW 1=sB0/4ped¹W B1

3¹W s1/Zndd, which was ignored by Shukla and Varma.27

Let us now consider the Alfvén waves in Eq.sA7d in thelimit of v@Vr0. Setting the first term on the left-hand sideof Eq. sA7d equal to zero we calculate the correction,D, tothe Alfvén frequency due to the presence of dust. We substi-tute v=kzVA+D and obtain

D = − Vr02 kz

2ky2VA

−Vr0

2

2ky2VA

2 kyVn = − Vr02 kz

2ky2VA

+ VSV/2.

sA9d

While the existence of the second term of Eq.sA9d wasknown and shown to be necessary for the weak turbulence ofshear Alfvén waves in an inhomogeneous plasma,26 the ex-istence of the additional frequency shift represented by thefirst term of Eq.sA9d was pointed out in paper 1. There wealso showed that this frequency shift is important because itcan lead to weak turbulence in a homogeneous plasma forshort wavelengths. The relative importance of the two termsin Eq. sA9d can be assessed from their ratio. We find that theratio R of the second and first terms depends only on theparameters of the medium, i.e.,

R=ld

Znd

dsZndddx

= S mic2ni

4pe2Z2nd2D1/2d lnsZndd

dx, sA10d

where we assumeky,kz.In the dense molecular clouds we estimatedld as,1 a.u.

and the visible size of the cloud structures as,100 a.u. andconclude thatR!1. In the inside and outside of heliospherethe dust mass density is estimated as a few percents of hy-drogen plasma density. When crossing heliosphere the Ul-ysses spacecraft had measured about 1000 impact eventswith grains.29 This corresponds to a few grains in 106 m3. Ifwe assume that the ambient plasma density is,0.1/cc weget ld,2 a.u. andR,1. So, we can conclude that for thedense molecular clouds the first term in Eq.sA9d is dominantwhile in the heliosphere both terms are important and shouldbe taken into account when considering the Kolmogorov-type cascade spectra.26

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042111-9 Magnetodynamics of a multicomponent ~dusty! plasma. II.... Phys. Plasmas 12, 042111 ~2005!

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