Nonlinear waves in collisional dusty plasma

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Nonlinear waves in collisional dusty plasma

B. P. Pandey, S. V. Vladimirov, and A. Samarian

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Nonlinear waves in collisional dusty plasma

B. P. Pandey,1,2,a S. V. Vladimirov,2,b and A. Samarian2,c1Department of Physics, Macquarie University, Sydney, NSW 2109, Australia

2School of Physics, The University of Sydney, NSW 2006, Australia

Received 22 October 2007; accepted 9 April 2008; published online 29 May 2008

The nonlinear wave propagation in collisional dusty plasma is investigated in the present work.

When the electrons and ions are highly magnetized, i.e., when the Lorentz force acting on the

plasma particles dominates its collision with the dust and charged dust remains weakly magnetized,

the drift between the plasma and dust particles may significantly modify the wave characteristics of

the medium. It is shown that when electrons and ions move approximately with the same bulk

velocity, the large amplitude waves can be easily excited in such a collisional dusty medium and

they can be described by derivative nonlinear Schrdinger equation. It is quite possible that the

soliton solutions of the nonlinear equation may be useful in explaining the parsec scale structures in

the astrophysical plasmas. 2008 American Institute of Physics.DOI:10.1063/1.2918341

I. INTRODUCTION

Alfvn waves area possible source of turbulence in the

interstellar medium.1

These waves are responsible for the

transport of angular momentum and energy in the accretion

disks.2

When the amplitude of the waves is large, a nonlinear

effect plays an important role in the propagation of these

waves. It is known that in a homogeneous, uniform back-

ground medium, the interplay between dispersion and non-

linearity can give rise to solitary wave structures.321

The

standard magnetohydrodynamic MHD waves have degen-eracy; i.e., two wave propagation velocities coincide when

the wave normal direction is parallel to the ambient back-

ground magnetic field. For fast modes when the sound ve-locity is less than the Alfvn velocity, nonlinearKortewegde Vries equation can be derived for an oblique

wave normal case,5,6

whereas for intermediate waves, modi-fied Kortewegde VriesMKdV can be derived.7 For nearlymonochromatic and strongly dispersive waves, nonlinear

Schrdinger equation NLSE can be derived.810 In a widevariety of space plasmas, NLSE describes nonlinear

electrostatic11,12

as well as electromagnetic13

structures.

When the wave degeneracy is retained and the wave normal

is quasi-parallel to the ambient magnetic field, instead of two

separate KdV and MKdV equations, two coupled equations

for the confluent MHD modes can be combined to give de-

rivative nonlinear SchrdingerDNLSequation.4 The NLSEcan be obtained from DNLS in the in the limit of long wave-

length modulation of the plane waves.20

Such waves have

been studied for last several decades in space plasmas.321

Most of the space plasma such as those in cometary tails,

interstellar molecular clouds, and planetary nebulae is

dusty.2226

It contains charged grains and, the coupling of the

grains to the magnetic field determines thewave propagation

in the planetary and interstellar medium.2232

The plasma-

grain collision can causes not only the damping of the high

frequency waves but also assist the excitation and propaga-

tion of the low frequency fluctuations in the medium. For

example, if the plasma-dust collision frequency is higher

than the dynamical frequency then collision will be respon-

sible for dragging the dust along the plasma fluctuations. In

such a scenario collision will always cause the propagation

of the waves in the medium without any damping. In the

opposite case, for high frequency oscillations, the dust-

plasma collision will cause the damping of the wave.

The large scale structures such as clumps and filaments

often provide the initial conditions in many astrophysical

environments. Since the propagation of low frequency fluc-

tuations in a collisional medium is possible, it should be

possible to investigate properties of nonlinear wave in such a

medium as such an investigation can be utilized to explain

the highly nonlinear structures in the collisional dusty envi-

ronment. Although the motivation of this work is to apply theresults to the space plasmas, the role of neutral is completely

neglected. Furthermore, the large dispersion in mass, charge,

and size of the grain, which can significantly influence the

propagation of the waves, will be neglected.

The dusty plasma dynamics can be studied in either of

the two limits: i The heavy dust particles provide a sta-tionary background and couples to the plasma fluid through

the charge neutrality condition, and ii the perturbations areon the order of or less than the typical plasma frequencies of

the dusty fluid. The Alfvn wave in the presence of immobile

dusty background have been studied by several authors.3033

The magnetized dusty medium can excite pair of circularlypolarized Alfvn waves.24,26,27,3436

The parametric instabil-

ity of a finite amplitude, circularly polarized Alfvn wave in

a dusty medium have been studied in the space and astro-

physical plasmas.3437

It is known that the collisional pro-

cesses in a weakly ionized dusty plasma can strongly influ-

ence the nonlinear ambipolar diffusion and the ensuing

current sheet formation.37

Further collision between plasma

particles and the grain is responsible for some of the novel

features in dusty plasma. As an example, the new collective

behavior is known to exist in such a plasma due to the charge

aElectronic mail: [email protected].

bElectronic mail: [email protected].

cElectronic mail: [email protected].

PHYSICS OF PLASMAS 15, 0537052008

1070-664X/2008/155/053705/5/$23.00 2008 American Institute of Physics15, 053705-1


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fluctuations: an offshoot of collision.26,28,38,39

Recently, the

parametric instability of a collisional dusty plasma have been

investigated.35,36

The present work investigates the nonlinear wave prop-

erties of collisional dusty plasma consisting of electrons,

ions, and charged grains. It is shown that the collisional

dusty medium is inherently dispersive in nature and the bal-

ance between the dispersion and nonlinearity leads to DNLS

equation. We shall assume that the charge on the dust particleis constant. Generally grain charge is a function of the

plasma parameter.38,39

However, we shall assume that the

charge fluctuation time is much larger than all the relevant

time scales in this work. The basic set of equations is dis-

cussed in Sec. II. It is shown that when the electrons and ions

are magnetized, the relative drift between the charged grains

and the plasma particleselectrons and ionsgives rise to theHall diffusion in the medium. This Hall diffusion causes the

wave dispersion. In many dusty plasma situations, Hall

MHD is the only proper description of the dynamics.32

In

Sec. III, the method of reductive perturbation, DNLS is de-

rived. In Sec. IV, a discussion of the results is given and the

results are summarized.

II. BASIC MODEL

The simplest description of dusty plasma, consisting of

the electrons, ions, and charged grains is given in terms of

continuity and momentum equations for respective species

with a suitable closure model; viz., an equation of state. The

continuity equation is

j

t+jvj= 0 . 1

Here, j is the mass density, vj is the velocity, and j stands

for electrons, ions, and grains. The momentum equations are

0 = PeeneE+ ve Bc eedve, 2

0 = Pi+eniE+ vi Bc iidvi, 3

ddvd

dt= Pd+Zen dE+

j=e,i

jjdvj. 4

Here, E

=E+vdB / c is the electric field in the dust framewith E and B as the electric and magnetic fields, respec-

tively,e is the electric charge, Zis the number of charge on

the grain, nj is the number density, and jd= ndvjd is thecollision frequency of the dust with jth species. Equations

24 on the right-hand side have a pressure gradient term,Lorentz force term, and collisional momentum exchange

terms. The electron and ion inertia have been neglected in

Eqs.2and 3.This is motivated by the fact that the inertiaof a dusty plasma is largely due to the presence of the

micrometer-sized dust grains. The neglect of plasma inertia

terms implies that we are primarily interested in the very low

frequency modes; i.e., id. Therefore, we shall anticipate

that these low frequency waves in the dusty medium will

propagate undamped.

In the presence of very small grains like MathisRumpl

Nordsieck distributions40

or polycyclic aromatic hydro-

carbons,41

when the grain mass is four to five orders of mag-

nitude larger than the ions, ion inertia may become as impor-

tant as the grain inertia and present formulation may need to

be generalized. The momentum equations 24 are closed

by assuming an isothermal equation of state Pj = Cs2

j, whereCs =Tj / mj is the sound speed.

We shall define mass density of the bulk fluid as =e+i +dd. The bulk velocity is, then, v=ivi +eve+dvd /vd. The continuity equation summing up Eq.1 for the bulk fluid becomes

t+v= 0 . 5

The momentum equation can be derived by adding Eqs. 2and4,

dv

dt = P+

J B

c . 6

Here, P = Pe + Pi + Pd is the total plasma pressure and J

= eneve + enivi is the current density in the dust frame. It is

interesting to note that the collision terms have disappeared

from the above momentum equation6.It should be pointedout that if one is interested in the high frequency oscillations

then the plasma inertia term may become important as well

and the present single-fluid momentum equation 6 will notbe valid.

In order to derive an induction equation, we note that

Eqs.2 and3 can be inverted to yield

vi =

c

BiE Pieni + i

2 cE B

B2 i

Pi Biid

1 + i2

,

7

ve =

c

BeE + Pe

ene+ e2 cE B

B2 + e

Pe B

eed

1 + e2

,

where B=B /B and

vi= icE

BPi

iid, ve= e

cE

BPe

eed. 8

Here,j =cj /jdis the plasma Hall parameter and is a mea-

sure of magnetization of the electrons and ions. For example,

when plasma cyclotron frequency cj dominates the plasma-

dust collision frequency, i.e., e1, and i1, the Hall

term EB will dominate in Eq. 7. Physically, in thedust frame, j1 implies that the plasma drift due to colli-

sion is very small compared to the transverse gyration of the

plasma particles across the magnetic field. This results lo-

cally in the plasma particles going away or coming close to a

stationary observer in the dust frame resulting in a time-

dependent Hall electric field. This Hall field is generated

over plasma-cyclotron time scale and depending upon the

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sign of the grain charge, the Hall scale can become arbi-

trarily large.32

In j1 limit, one may assume that the relative drift

between electrons and ions are small, i.e., vevi, and theelectron velocity ve can be written as

ve= J

Zend. 9

While writing Eq. 9, plasma quasi-neutrality condition ne= ni +Zndhas been used. After taking the curl of Eq. 2,i.e.,E+veB /c0, and making use of Maxwells equation, theinduction equation can be written as

B

t= vd BJ B

Zend . 10

We see from the induction equation10that the ideal-MHDlimit correspond to B /Zend0. Note that the ideal-

MHD description of the two-component plasma assumes that

the relative drift between electrons and ions are absent; i.e.,vevi. The reason for the Hall effect in such a plasma isattributed to the finite ion inertiawhich breaks the symmetrybetween electrons and ions with respect to the magnetic

field. This symmetry breaking results in the finite relativedrift between the electrons and ions, and thus E+veB / c

=0, is written as E+viB / c =JB / ene, since ve =viJ / ene. However, in a dusty plasma, or for that matter in any

multicomponent plasma, since neni, even when relative

drift between the plasma particles are absent, i.e., ve =vi, ow-

ing to the presence of a third, charged component, the Hall

effect will always be present.32

Whenj1, i.e., collisional terms dominate the Lorentz

force terms in the momentum equation, and from Eq. 7,veecE /B and viicE /B. The induction equationthen becomes

B

t= vd Beff B , 11

where

eff=c2Ohm

41 ie

Znd

ne

, 12

and

Ohm =meed

nee2

. 13

In the expression for eff, we have neglected the term on the

order ofi /e in the denominator. It is clear from Eq. 11that in the collisional dominated regime, magnetic flux de-

cays due to Ohmic dissipation. When Znd/nee /i, thedissipative term can become very large and magnetic field

evolution can be solely dictated by the Ohmic diffusion in

such a medium.

We investigate the dusty plasma dynamics in the j1

limit with the help of Eqs. 5, 6, and 10 along with anequation of state P /=const.

III. THE DNLS EQUATION

Assume a uniform background field in the z direction

and that the perturbations are one-dimensional with thevariation along the z direction. The above set of Eqs. 5,6,and 10 in the stretched variable =z VAt, =

2t can

then be written as

D+vz

= 0 ,

D+ vz

vz= p+ B2

8 ,

14

D+ vz v =Bz

4

B

,

DB =Bzv

vzB+ VAdz

0

B

.

Here, D =/ VA/, VA2 =Bz

2/ 40, d= VA /cd is the

dust skin depth, =1, and v and B are the transverse

to z components of velocity and magnetic field, respec-tively.

Equations 14 admit monochromatic circularly polar-

ized waves Bx+ iBy =B0expit kz, vz = 0, =0 as an ex-act solution. For a forward propagating wave k0, thefollowing dispersion relation is satisfied:

kVA= 0.5kd+1 + 0.25kvAkd/cd, 15

where =+1 and =1 will correspond to right and left

circularly polarized waves, respectively.

Recently, DNLS has been derived using reductive per-

turbation method in infinitely conducting plasma and it has

been shown that solitons propagating at an arbitrary angle

with respect to the ambient magnetic field may exist insuch a plasma.

17We follow similar approach in the present

work and assume that the perturbations to physical quantities

vanish at infinity; i.e., 0, pp0, vz0, and BB0. We shall use inverse of plasma = cs

2/VA

2 , where cs2

=p0 /0 is the acoustic speed as the smallness parameter;i.e., =1 for perturbative expansion of the physical quan-

tities. Introducing scaled sound speed cs =cs and the scaled

equilibrium pressurep0 =p0, the scaled pressure is expanded

as p =1p0 +p1 +p2 +. We note that p1, p2 are scaled

pressure here. Expanding all other quantities as f=f1 +f2+, from Eqs. 14 the following terms of the order 0 areobtained:

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VA1

1vz1

= 0 ,

1VA vz1vz1

=

p1+ B1

2

8 ,

1VA vz1v1

= B

z

4B1

, 16

VAB1

= Bz

v1

+

vz1B1,

p0

1

=p0

0

.

The pressure density relation in Eq.16has been derived byutilizing boundary condition at infinity. We see that

1

=0

.

We can perform the integration of the above equations and

eliminate the constants by utilizing the boundary conditions

at infinity. Then vz1 =0 and

v1= VAB1B0

Bz, p1=

B12 B0

2

8.

17

The next order terms are

VA2

0vz2

= 0 ,

0VAvz2

=

p2+ B1 B2

4 ,

18

0v1

VAv2

= Bz

4

B2

,

B1

VA

B2

=

Bzv2 vz2B1+VAdz

2

B1

2 ,

and p1 = cs22. Making use ofp1 from Eq.17 and eliminat-

ing2in favor ofp 1in the first equation of Eqs.18, we canwrite

vz2= VA

0cs2

B12 B0

2

8. 19

Using v1 from Eq.17,the last two equations of Eqs.18can be written as

VAB2

+Bz

v2

=

B1

,

20

VAB2

+Bz

v2

=

B1

+ 2Dz

2B1

2

2

B1B1

2 B02 ,

where= VA / 160cs2, and D = VAd/2. Making use of the

first equation of Eqs. 20 in the second one leads to thefollowing DNLS equation:

B1

B1B1

2 B02+Dz

2B1

2 = 0 .

21

By writing above equation in the component form in the x

and y directions, and combining them into a single complex

variable b =Bx+ iBy, one gets the complex scalar version of

DNLS,

b

bb2 b0

2+iD

2b

2= 0 . 22

The DNLS equation 22 describes the evolution of weaklynonlinear, weakly dispersive waves propagating either ex-

actly parallel to the ambient magnetic field B=Bz corre-sponding to the boundary condition b00 as orslightly oblique propagationb00 at infinity. Dispersion inthe DNLS equation 22 is caused by the dust inertia andnonlinear term arises due to coupling between the transverse

magnetic field and plasma pressure perturbations. We note

that the derivation of the above DNLS equation is very simi-

lar to the MHD case although we have considered a set ofdissipative multicomponent dusty plasma equations. In the

low-frequency limit i.e., frequencies much lower than theplasma-dust collision frequencies, the dust-plasma collisioncauses the dust and the plasma fluids to stick together as a

single fluid. This results in a MHD-like set of equations with

dispersion due to Hall term: a manifestation of the multicom-

ponent nature of the plasma.

IV. DISCUSSION

For exactly parallel b0 = 0, circularly polarized, finiteamplitude Alfvn waves, b =B0exp i k satisfies thefollowing nonlinear dispersion relation:

= B02k+Dk2. 23

We note that when =0, Eq.23 is a linear dispersion rela-tion describing the whistler mode. When 0, for the left-

hand polarized waves = 1, k0 is unstable to parallelmodulation while the right-hand polarized branch = 1, k0, is stable to the parallel modulation.

4

For exactly parallel, circularly polarized, finite amplitude

Alfvn waves, Eq. 22 admits localized envelope solitonsolutions

20

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b2 =2V

2cosh V

L 11 , 24

where Vrepresents the velocity in the co-moving frame, L

=D / VVA / 2Vd is the width of the soliton, and 2V/ isthe maximum amplitude of the soliton. Here we have as-

sumed=1. It is instructive to calculate the soliton width for

the astrophysical plasmas. For example, in dense molecular

cloud coresthe neutral density 1010 cm3, charged grainsare more numerous than the electrons and ions.

23,24The ion-

ization fraction of the plasma is strongly affected by the

abundance and size distribution of the grains through the

recombination process on the grain surface. The ratio of dust

to neutral mass density is d/n =0.01. Although the grain

density is small in comparison with the neutral hydrogen

density in such clouds, it may significantly affect the dynam-

ics of star formation.2,2126

The grain size in such clouds may

vary from a few micrometers down to a few tens of atoms in

the interstellar medium. In fact, the presence of very small

grains polycyclic aromatic hydrocarbons can play an impor-tant role in the dynamics. The small grains are numerous and

their charge may vary between 1 and 0.23,24

Thus, if one

assumes md= 1015 g, then for mn = mp, nd= 10

11nn. Assum-

ing a dense molecular cloud with nn = 106 cm3, one gets nd

= 105 cm3. For a typical mGauss field B = 103 G, one gets

VA0.1 km and dust skin depth d105 km. Thus, L

108 pc, where 1 pc31013 km. Therefore, the width ofthe nonlinear structure is too small to be meaningful in cloud

cores of a few pc. Given the uncertain nature of various

numbers used here, one can arrive at entirely different con-

clusion by choosing different numbers. For example, a much

heavier dust md1012

g will give VA1 km and dustskin depth d1011 km. Thus, L0.01 pc, which is compa-

rable to the size of the dense cloud cores. Therefore, it is

tempting to conclude that the soliton may explain some of

the observed structures in the astrophysical plasmas. How-

ever, given the uncertain nature of numerical parameters,

such a conclusion must be treated with fair degree of caution.

One should also note that the present formulation does not

include the neutral dynamics that dominates the clouds.

To summarize, we have shown that in the low frequency

limit, when the electrons and ions are highly magnetized, the

multifluid set of equations reduces to the set of equations

which are very similar to the single-fluid Hall MHD equa-

tions. The nonlinear propagation of the waves in the magne-

tized, collisional dusty medium can be described by the

DNLS. The soliton solution of DNLS could be probably in-

voked to explain the pc range structures in the molecular

cloud.

ACKNOWLEDGMENTS

The support of the Australian Research Council is grate-

fully acknowledged for the present work.

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053705-5 Nonlinear waves in collisional dusty plasma Phys. Plasmas 15, 053705 2008


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Nonlinear waves in collisional dusty plasma

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