# Response to ''Comment on 'Instability of the Shukla mode in a dusty ...

Response to ''Comment on 'Instability of the Shukla mode in a dusty ...

Response to ''Comment on 'Instability of the Shukla mode in a dusty ...

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PHYSICS OF PLASMAS VOLUME 11, NUMBER 8 AUGUST 2004

**on**se**on** ‘Instability **of** **the** **Shukla** **mode** **in** a **dusty**

plasma c**on**ta**in****in**g equilibrium density and magnetic field **in**homogeneities’

and ‘New res**on**ance and cu

waves **in** **dusty** magne

P. K. **Shukla**, a) R. Bharuthram, b) A. A. Mamun, c) G. E. Morfill, d) R. Schlickeiser,

and L. Stenflo e)

Institut für Theoretische Physik IV and Centre for Plasma Science and Astrophysics, Fakultät für Physik

und Astr**on**omie, Ruhr-Universität Bochum, D-44780 Bochum, Germany

Received 5 May 2004; accepted 18 May 2004; published **on**l**in**e 20 July 2004

It is shown that **the** Comment **of** Rudakov **on** **the** papers by **Shukla** et al. and Mamun et al. is

mislead**in**g. © 2004 American Institute **of** Physics. DOI: 10.1063/1.1770845

The purpose **of** this **on**se**in**t out that

Rudakov 1 is not justified **the** work **of** **Shukla**

et al. 2 and Mamun et al. 3 We shall **the**refore dem**on**strate that

his criticisms are irrelevant.

Rudakov 1 claims that **Shukla** et al. 2 have revisited **the**

magnetic drift wave **in**stability, a

years ago **in** **the** c**on**text **of** electr**on** magne

EMHD, 4,5 and **the**n **in** **dusty** plasma physics. 6 We refute this

claim **on** **the** grounds given below.

The EMHD describes wave phenomena magnetic electr**on**

drift waves 5 and obliquely propagat**in**g electr**on**

whistlers 7 **on** time scales much l**on**ger than **the** i**on** plasma

and i**on** gyroperiods. Thus, i**on**s form a neutraliz**in**g background

and **the** i**on** fluid velocity is zero. In such a situati**on**,

**the** n**on**l**in**ear wave moti**on**s **in** a collisi**on**less, cold electr**on**

plasma is governed by 7

t

B c2

2 2

pe

B

“ v e

B c2

2 2 B0, 1

pe

where B is **the** magnetic field, c is **the** speed **of** light **in**

vacuum,

v e

c

“B

2

4en e

is **the** electr**on** fluid velocity, pe (4n e e 2 /m e ) 1/2 is **the**

electr**on** plasma frequency, n e is **the** electr**on** number density,

e is **the** magnitude **of** **the** electr**on** charge, and m e is **the**

electr**on** mass. Equati**on** 1 admits obliquely propagat**in**g

electr**on** whistlers which have **the** frequency

**in** a uniform magne**the** wave number,

is **the** angle between **the** wave vec**the** external

magnetic field B 0 ẑ, ẑ is **the** unit vec**on**g **the** z axis,

ce eB 0 /m e c is **the** electr**on** gyr**of**requency, and p is **the**

unperturbed electr**on** plasma frequency. In a n**on**uniform

magne**on**e can also have **the** compressi**on**al magnetic

electr**on** drift **mode** k y cB 0 /4en e L n (1k 2 2 e ), as

reported **in** Ref. 5. Here, k y is **the** y comp**on**ent **of** **the** wave

vec**the** electr**on** density gradient scale length, and

e c/ p is **the** electr**on** sk**in** depth. However, **the** magnetic

drift waves **in** an electr**on** plasma are quite different from **the**

**Shukla** **mode**, 8 which exists **in** an electr**on**-i**on** plasma c**on**ta**in****in**g

stati**on**ary charged dust gra**in**s. The **Shukla** **mode** is a

low-frequency **in** comparis**on** with **the** Rao cut-**of**f

frequency 9 compressi**on**al dispersive wave **in** a n**on**uniform

magne**in** Ref. 6. In his

present revised Comment, 1 Rudakov suggests that **the** frequency

**in** Ref. 6 is k y cB 0 /4eZ d n d L nd (1k 2 2 d ),

where Z d is **the** number **of** charges resid**in**g **on** **the** dust gra**in**,

n d is **the** dust number density, L 1 nd d ln(Z d n d )/dx is **the** **in**verse

scale length **of** **the** dust density gradient, d

d i n i /Z d n d , d i c/ pi is **the** i**on** sk**in** depth, and pi is **the**

unperturbed i**on** plasma frequency. However, it should be

noted that **in** **the** above Rudakov is just copy**in**g down **the**

**Shukla** **mode** frequency, 8 as **the** latter cannot be derived from

Eq. 18 **in** Ref. 6 which does not c**on**ta**in** **the** density **in**homogeneity

term note that **the** third term **in** **the** left-hand side

**of** Eq. 18 is n**on**l**in**ear s**in**ce bB(x,y)/B 0 ]. Fur**the**rmore,

Ref. 6 assumes that **the** wave frequency is much smaller

than **the** i**on** gyr**of**requency ci , which is a significantly different

c**on**diti**on** than that **of** Ref. 8 which supposes that

Z d n d ci /n e . Rudakov 1 claims that **in** Ref. 6 **the** dust comp**on**ent

is unmagnetized because **of** **in**ertia and collisi**on**s

with a neutral gas, but **the** electr**on**s and i**on**s are magnetized.

In **the** **Shukla** **mode**, both electr**on**s and i**on**s are magnetized

and **the** dust gra**in**s are stati**on**ary. Clearly, Rudakov is mislead**in**g

**the** readers.

The **Shukla** **mode** 8 **in** a n**on**uniform **dusty** magne

Also at **the** Department **of** Physics, Umeå University, SE-90187 Umeå,

Sweden, as well as at **the** Center for Interdiscipl**in**ary Plasma Science,

Max-Planck-Institut für Extraterrestrische Physik und Max-Planck-Institut

für Plasmaphysik, D-45740 Garch**in**g, Germany, and GoLP/Institu

Técnico, Universidade Técnica de Lisboa, Lisboa 1049-001, Portugal.

b Permanent address: Department **of** Physics, University **of** Natal, 4041 Durban,

South Africa.

c Permanent address: Department **of** Physics, Jahangirnagar University, Savar,

Dhaka, Bangladesh.

d Permanent address: Max-Planck-Institut für Extraterrestrische Physik,

D-45740 Garch**in**g, Germany.

e Permanent address: Department **of** Physics, Umeå University, SE-

90187 Umeå, Sweden.

k2 c 2 ce cos

p 2 k 2 c 2

3

1070-664X/2004/11(8)/4156/3/$22.00

4156

© 2004 American Institute **of** Physics

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Phys. Plasmas, Vol. 11, No. 8, August 2004 **on**se**on** ‘Instability **of** **the** **Shukla** **mode** ...

4157

plasma with immobile charged dust gra**in**s is based **on** **the**

modified Hall-magne**on**s

composed **of** **the** i**on** c**on**t**in**uity equati**on**, Faraday’s law

B

t “ n i

n e

v i c“B

4en eB,

and Rao’s modified i**on** momentum equati**on** cf. Eq. 2 **in**

Ref. 9

t

i•“ v v i R v i ẑ “BB

5

4n e m i

**in** **the** low-frequency limit (/t)v i •“ **the** magnitude

**of** **the** Rao cu

is **the** dust charge equals Z d e for negatively charged dust

gra**in**s and Z d e for positively charged, ci eB 0 /m i c is **the**

i**on** gyr**of**requency, and m i is **the** i**on** mass. It should be

stressed that Eq. 4, which c**on**ta**in**s **the** Coriolis-like force

R v i ẑ due **the** separati**on** **of** charges **in** **the** presence **of**

stati**on**ary charged dust gra**in**s, has been obta**in**ed by substitut**in**g

**the** electric field from **the** **in**ertialess electr**on** momentum

equati**on** **in****the** usual i**on** momentum equati**on**, and

**the**n by elim**in**at**in**g **the** electr**on** fluid velocity by us**in**g Ampère’s

law and **the** quas**in**eutrality c**on**diti**on** en i en e

q d n d . Thus, **the** effect **of** immobile charged dust **in** Eqs.

4 and 5 appears through **the** latter as n i n e **in** **dusty**

plasmas. Fur**the**rmore, c**on**trary **the** asserti**on** **of** Ref. 1, we

emphasize that Eq. 4 was not obta**in**ed by Smith and

Brice 10 who c**on**sidered **the** propagati**on** **of** electromagnetic

waves **in** a multi-i**on** magne**on**ta**in****in**g electr**on**s

and multiple positive i**on**s, and showed that for each i**on**ic

species for waves propagat**in**g perpendicular **the** magnetic

field, **the**re exists a multiple-i**on** res**on**ance and a multiple-i**on**

cu**on** magne**on**e **of** **the** i**on** species

cannot be assumed immobile due **the** narrow range **of** i**on**

charge **on** mass distributi**on**s. In **dusty** plasmas, this separati**on**

is allowed as Z d /m d Z i /m i , and **the** dust gra**in**s can

be assumed stati**on**ary as d**on**e **in** deriv**in**g Eqs. 4 and 5.

Here, m d is **the** dust mass and Z i is **the** i**on** charge state.

Mamun et al. 11 have presented a new cut-**of**f frequency for

low-frequency electromagnetic waves **in** a multi-i**on** magne

with stati**on**ary charged dust gra**in**s.

**Shukla** et al. 2 have c**on**sidered **the** **in**stability **of** **the** flutelike

**Shukla** **mode** **in** **the** presence **of** equilibrium density and

magnetic field **in**homogeneities. They have presented a selfc**on**sistent

plasma equilibrium **in** crossed dc electric and

magnetic fields. Their equati**on**s 3 and 4 relate **the** dc

electric field with a c**on**stant magnetic field gradient **in** a

self-c**on**sistent manner. Such an equilibrium is absent **in** **the**

present work **of** Rudakov as well as **in** Ref. 6. Fur**the**rmore,

**Shukla** et al. 2 have shown that **the** magnetic field **in**homogeneity

provides a l**in**ear coupl**in**g between **the** compressi**on**al

magnetic field and **the** electr**on** density perturbati**on**, see, for

example, Eqs. 11–13 **in** Ref. 2. In **the** local approximati**on**,

which holds for wavelengths much smaller than **the**

scale lengths **of** **the** density and magnetic field **in**homogeneities,

**Shukla** et al. 2 found **the**ir Eqs. 15 and 16, and that

**the** equilibrium plasma is unstable aga**in**st **the** modified

**Shukla** **mode** provided that **the** equilibrium magnetic field

4

and plasma density gradients oppose each o**the**r. Physically,

**in**stability arises because **of** a f**in**ite value **of** **the** divergence

**of** **the** product **of** **the** density perturbati**on** and **the** equilibrium

i**on** drift **in** an **in**homogeneous magnetic field. The electr**on**

density and magnetic field gradients **in** comb**in**ati**on** with **the**

sheared wave electric field produce an electr**on** flux, which

breaks **the** frozen-**in**-field l**in**es. The result**in**g phase shift between

**the** density and compressi**on**al magnetic field perturbati**on**s,

**in** turn, produces **in**stability **of** **the** modified dispersive

**Shukla** **mode**, governed by Eq. 15 **in** Ref. 2 which

c**on**ta**in**s **the** dispersive term k 2 2 , where **the** modified i**on**

sk**in** depth d differs from i . We thus do not need **the**

**in**clusi**on** **of** **the** dust dynamics for produc**in**g **in**stability, c**on**trary

**the** suggesti**on** **of** Rudakov. 1

Rudakov’s analysis **of** a n**on**dispersive compressi**on**al

**mode**, given by his Eq. 4, is based **on** **in**ertialess electr**on**s

and i**on**s e.g., Eqs. 1 **in** Ref. 1, c**on**trary **the** modified

**Shukla** **mode** 2 which employs **in**ertialess electr**on**s but **in**ertial

magnetized i**on**s **in** **the** low-frequency limit **in** which **the**

i**on** fluid velocity is c**on**trolled by **the** Coriolis-like force **of**

Rao **the** first term **in** **the** right-hand side **of** our Eq. 4

above, which is dependent **on** **the** presence **of** immobile

charged dust gra**in**s **in** an electr**on**-i**on** plasma. Hence, Rudakov’s

stability argument based **on** his Eq. 3 is useless, s**in**ce

**the** situati**on** described **in** Ref. 2 is quite different from what

Rudakov presents **in** his Comment. 1

Incidentally, **the** l**in**ear and n**on**l**in**ear low-frequency **in**

comparis**on** with **the** i**on** gyr**of**requency electromagnetic

waves **in** a **dusty** magne**in**s and

**in**ertialess electr**on**s and i**on**s are described by **the** dust

H-MHD equati**on**s 12 c**on**sist**in**g **of** **the** dust c**on**t**in**uity equati**on**

d

t “• dv d 0,

6

**the** dust momentum equati**on**

t

d•“ v v d “BB

, 7

4 d

and Faraday’s law

B

t “ v d c“B

4q d n d B,

where d n d m d is **the** dust mass density. Equati**on**s 6–8

admit dust Alfvén waves **on** temporal and spatial scales that

are **of** **the** order **of** **the** dust gyroperiod and **the** dust sk**in**

depth. Thus, our Eqs. 6–8 above are much more appropriate

for wave studies than Eqs. 1–3 **of** Rudakov, 1 which

completely neglect **the** dust fluid velocity **in** **the** **in**ducti**on**

equati**on** 3.

We now turn our attenti**on** **the** work **of** Mamun et al. 3

who presented **the** dispersi**on** properties **of** low-frequency **in**

comparis**on** with **the** i**on** gyr**of**requency dustelectromagnetic

waves **in** a four comp**on**ent uniform magne

Their **mode**l c**on**sists **of** **in**ertialess cold electr**on**s

and i**on**s e.g., Eqs. 1 and 2 **in** Ref. 3 and two comp**on**ents

**of** **in**ertial and magnetized charged dust gra**in**s, which

obey Eqs. 3 and 4 **in** Ref. 3. They showed that **the** simul-

8

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4158 Phys. Plasmas, Vol. 11, No. 8, August 2004 **Shukla** et al.

taneous presence **of** positive and negative **in**ertial dust gra**in**s

**in**troduces a new res**on**ance and cu

waves, and a sharp forbidden-frequencyband

**in** between **the** two branches **of** **the** propagat**in**g dustelectromagnetic

waves. This is a completely different

scenario as compared **of** **the** Buchsbaum’s res**on**ance 13

**in** a magnetized plasma with two i**on** species without charged

dust gra**in**s. Buchsbaum found that **the** presence **of** two-i**on**

species **in**troduces an additi**on**al res**on**ance for electrostatic

waves propagat**in**g across **the** static magnetic field at a frequency

between **the** two-i**on**ic gyr**of**requencies. Hence, **the**

work **of** Mamun et al. 3 and that **of** Buchsbaum 13 cannot be

compared, and Eqs. 6 and 7 **of** Rudakov 1 are irrelevant as

**the**y do not **in**corporate **the** **dusty** plasma **mode**l **of** Mamun

et al. 3 We stress that nei**the**r Smith and Brice 10 nor

Buchsbaum 13 have charged dust gra**in**s **in** **the**ir descripti**on**s.

F**in**ally, it is **in**explicable that Rudakov goes **on** suggest**in**g

that **the** wavelength **of** **the** dust Alfvén waves should be **of**

**the** order **of** 10 km for labora**on**diti**on**s, which is

**the**ory **of** Mamun et al. 3 would work

**in** magnetized labora**dusty** discharges for waves with k

pd /c. The latter is satisfied for measurable waves **in**

magnetized **dusty** plasmas.

In c**on**clusi**on**, **the** criticism **of** Rudakov **in** Refs. 2 and 3

is peculiar, as it does not c**on**ta**in** relevant **in**formati**on**. The

materials presented **in** Ref. 1 as well as **the** his**on**s

are mislead**in**g **the** readers **of** Physics **of** Plasmas.

ACKNOWLEDGMENTS

This work was partially supported by **the** European

Commissi**on** through C**on**tract No. HPRN-CT-2000-00140 as

well as by **the** Deutsche Forschungsgeme**in**schaft through **the**

S**on**derforschungsbereich 591 and **the** Swedish Research

Council through **the** C**on**tract No. 621-2001-2274.

1 L. I. Rudakov, Phys. Plasmas 11, 41542004.

2 P. K. **Shukla**, R. Bharuthram, and R. Schlickeiser, Phys. Plasmas 11, 1732

2004.

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2004.

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6 L. I. Rudakov, Phys. Scr., T 89, 1582001.

7 P. K. **Shukla** and L. Stenflo, Phys. Lett. A 259, 491999.

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2003; Phys. Rev. E 69, 047401 2004.

9 N. N. Rao, J. Plasma Phys. 53, 317 1985.

10 R. L. Smith and N. Brice, J. Geophys. Res. 69, 5029 1964.

11 A. A. Mamun, P. K. **Shukla**, and G. E. Morfill, Phys. Lett. A 323, 105

2004.

12 P. K. **Shukla**, G. E. Morfill, and V. Krishan, Phys. Scr. T

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