Electrostatic modes and instabilities in nonideal dusty plasmas with sheared flows and grain charge fluctuations N. N. Rao Citation: Phys. Plasmas 6, 2349 (1999); doi: 10.1063/1.873506 View online: http://dx.doi.org/10.1063/1.873506 View Table of Contents: http://pop.aip.org/resource/1/PHPAEN/v6/i6 Published by the American Institute of Physics. Additional information on Phys. Plasmas Journal 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 Downloaded 13 Apr 2013 to 131.211.208.19. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://pop.aip.org/about/rights_and_permissions
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Electrostatic modes and instabilities in nonideal dusty plasmas withsheared flows and grain charge fluctuationsN. N. Rao Citation: Phys. Plasmas 6, 2349 (1999); doi: 10.1063/1.873506 View online: http://dx.doi.org/10.1063/1.873506 View Table of Contents: http://pop.aip.org/resource/1/PHPAEN/v6/i6 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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Waves and instabilities in ionized gases consisting,addition to electrons and ions, of finite-sized charged partlate matter in the sub- as well as super-micron ranges hbeen extensively investigated beginning in this decade.1–22
The importance of these studies arises from the fact thatis a ubiquitous component of matter both in laboratory,well as in space and astrophysical environments, and pladominant role in determining the collective behavior of ioized media such as plasmas in the low-frequency regiWhile in laboratory devices such as tokamaks, dust partican give rise to instabilities23 resulting in enhanced anomalous diffusion which is detrimental to confinement, the preence of trapped dust impurities leads to serious contamtion problems during the industrial manufacturesemiconductors by means of plasma-aided processes24–26
On the other hand, dust is more commonly found interrestrial surroundings and beyond, such as Earth’s mpause region, cometary atmospheres, planetary rings anterstellar clouds.27
There are three characteristic features that distinguthe so-called dusty plasmas from the usual multicomponplasmas, all of which owe their origin to the large size ahence, the mass of the dust grains. First, the dust grainsusually about 10–12 orders of magnitude heavier than
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ions, and hence the plasma, as well as the gyro-frequenof the dust grains, and the ions are widely separated. Coquently, it is possible to separate the normal modes arisdue to the ion and the dust dynamics. The latter gives risnew types of ultralow frequency waves, which are otherwnot possible in the usual electron-ion plasmas. For examthe first such normal mode arising solely due to dust dynaics was theoretically predicted by Raoet al.3 and named asthe ‘‘Dust–Acoustic Wave’’~DAW! wherein the inertia, aswell as the wave dynamics, is governed by the dust comnent, while the electron and ion thermal pressures provthe restoring force. Recent laboratory experiments on duplasmas have confirmed19–22 the existence of such modehaving frequencies in the range of about ten Hertz, andresults are in good agreement with the predictions ofDAW dispersion relation.3 Second, by virtue of their largesize, dust particles facilitate the flow of electron and icurrents on to their surface. The grain charge can fluctueither because of wave motion induced electron andnumber densities, or due to the equilibrium charging pcesses, and hence should be treated as a new dynamicaable. This leads to certain novel effects, which are absenthe usual multicomponent electron-ion plasmas. Forample, it is known8,18,28 that low-frequency waves such athe DAWs undergo damping due to grain charge flucttions, which would otherwise propagate as normal modThe damping rate is maximum when the charge fluctuattime scale is of the order of the wave period. Third, du
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2350 Phys. Plasmas, Vol. 6, No. 6, June 1999 N. N. Rao
plasmas exhibit nonideal effects arising due to the largeof the grains as well as high charging which is usually netive. The usual assumption of ideal behavior of the dust flis valid for grains in the sub-micron range and for dilugases when the characteristic potential energy due to Clomb interactions is much smaller than the mean therenergy. However, for grains in the super-micron rangeparticularly at high number densities, the interaction betwnear-neighbor particles as well as the volume reduccould be significant, giving rise to nonideal contributions15
For example, for grains with a radius of about 50 microthe volume reduction contribution is about 10% when thedensity is about 105 particles/cc. Such parameters are withthe range of present day experiments on dusty plasmas.
In the last few years, a number of authors have theorcally investigated the collective behavior of dusty plasmasterms of different types of waves and instabilities, manywhich do not exist in the usual electron-ion plasmas. Fexample, the mere presence of a charged dust compoleads to the existence of a low-frequency novel mode cathe dust ion-acoustic wave~DIAW !,4 which would have oth-erwise been heavily Landau damped in electron-ion plasmMagnetized dusty plasmas lead to the excitation of elecstatic modes such as the dust-cyclotron waves which cadriven unstable by means of external currents along the mnetic field,29,30 as well as a~parallel! Kelvin–Helmholtz~K–H! type of low-frequency instabilities,11,12,16 when thefield-aligned relative flow speed between adjacent layersceeds the DAW phase speed.3 Furthermore, as a possiblstationary state of the nonlinear evolution, the K–H instabity can give rise to the formation of vortex structures in duplasma.31 Inhomogeneous magnetized dusty plasmas supdifferent types of drift waves6,9 in the low- as well as high-frequency regimes. The presence of a charged dust comnent significantly modifies the range of unstable wave nuber for the occurrence of Rayleigh–Taylor instability industy plasma supported by the magnetic field agagravity.32 On the other hand, in the regime of electromanetic waves, low-frequency modes, such as the circularlylarized modes,5 dust-magnetoacoustic waves,33 and Alfvenwaves,34 have been studied in a dusty plasma embeddean external magnetic field. A novel feature of nonunifomagnetized dusty plasmas is the existence of finite frequeconvective cell modes which would not arise in a puelectron-ion plasma without the charged dust grains.34
While most of the existing analyses of wave propagatand instability excitation in dusty plasmas have focusedtention on the ideal behavior of the dust component, nonideffects in dusty plasmas have been discussed only in thecouple of years. In the weak coupling regime, the nonideffects can be incorporated by using an appropriate equaof state such as the van der Waals, which accounts forattractive cohesive forces between the grains as well asvolume reduction coefficient. Recently, we have presenelsewhere15,35 an analysis of DAW propagation in weaklnonideal plasmas by including the van der Waals equatiostate for the dust component. For the linear case, the volreduction coefficient enhances the phase speed of the DAwhile the molecular attractive forces lead to a decrease in
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phase speed, and the net contribution due to the two effdepends on relative values of the nonideality parametersthe nonlinear regime, supersonic DAW solitons are foundbe admissible in both the sub- as well as super-criticalrameters, whereas subsonic propagation near the DAW pspeed is possible only for the supercritical case. In the strcoupling regime, wave propagation in dusty plasmas habe studied using the framework of generalizhydrodynamics,36,37 while contributions due to dust–duscorrelation are described by the kinetic theory.38,39
We present in this paper an analysis of electrostamodes and instabilities in a weakly nonideal dusty plaswith sheared flows by including the effects of grain charfluctuations. Specifically, we investigate low-frequency eletrostatic modes such as the DAWs and the drift waves ininhomogeneous dusty magnetoplasma. Effects of nonidcontributions on the growth rate of density- as welltemperature-gradient driven~parallel! K–H instability havebeen analyzed. The plan of the paper is as follows: Innext section, the basic governing equations are presentedthe nature of the steady state equilibrium is discussedSec. III, we derive the generalized dispersion relation, apoint out a generic feature of dusty plasmas which supporzero-frequency, purely damped mode arising solely duethe grain charge fluctuations. Electrostatic mode propagais discussed in Sec. IV by explicitly deriving appropriadispersion relations, whereas Sec. V presents a discussiothe shear flow driven K–H instabilities. A summary of thmain results of the present work is given in Sec. VI.
II. BASIC EQUATIONS
We consider a magnetized dusty plasma having etrons, ions and dust grains embedded in a magnetic fialong the positivez-direction, B05B0z. We assume theplasma-b to be sufficiently small so that the magnetic fiefluctuation produced by the plasma motions is negligiband hence excited modes may be considered as essenelectrostatic. The present analysis is based essentially ondusty plasma model of Raoet al.3 wherein, for low-frequency modes, the electrons and the ions are assumbe in thermal equilibrium at their respective temperatu(Te and Ti) in the presence of self-consistent electrostapotential ~f! while the wave dynamics is governed by thheavier dust species. Accordingly, the electron (ne) and theion number (ni) densities are given by Boltzmann distributions
ne5ne0 expS ef
kBTeD , ~1!
ni5ni0 expS 2ef
kBTiD , ~2!
wheren0 j ( j 5e,i ) are the unperturbed number densities,kB
is the Boltzmann constant, ande is the magnitude of theelectronic charge.
The dust fluid is governed by the equations of continuand momentum balance
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2351Phys. Plasmas, Vol. 6, No. 6, June 1999 N. N. Rao
]nd
]t1¹•~ndvd!50, ~3!
ndmdF]vd
]t1~vd•¹!vdG
52ndqd¹f1ndqd
c~vd3B0!2¹pd , ~4!
together with the charge neutrality condition,qdnd1eni
2ene50. Here, nd , md , vd , qd and pd denote, respectively, the number density, the mass, the velocity, the chaand the pressure of the dust species. If the charging ofdust grains arises from plasma currents due to electronsions reaching the grain surface, the grain charge variableqd
is determined by the charge current balance equation
]qd
]t1~vd•¹!qd5I e1I i , ~5!
where the average electron (I e) and ion (I i) currents flowingonto the grain surface are determined by the respective lnumber densities as well as by the potential differencetween the grain surface and the ambient plasma. Assumthe streaming velocities of the electrons and the ions tomuch smaller than the respective thermal velocities (v te andv t i), we have the following expressions40,41 valid for spheri-cal grains of radiusR:
I e52peR2S 8kBTe
pmeD 1/2
ne expS ec
kBTeD , ~6!
I i5peR2S 8kBTi
pmiD 1/2
ni S 12ec
kBTiD , ~7!
wherec5qd /R denotes dust particle surface potential retive to the plasma potential, andme(mi) is the mass of theelectron~ion!. On the other hand, if the ion streaming veloity (v is) is much larger than the ion thermal velocity, thenapproximate expression for the ion current is given byI i
5pR2ev isni(122ec/miv is2 ), while for arbitrary values of
v is one needs to use a more complicated expression.42 Atequilibrium, the steady state floating potentialc0 and theequilibrium dust chargeqd0 are calculated from the currenbalance equation,I e1I i50, together withqd0[Cc0 whereC is the grain capacitance.
The system of Eqs.~1!–~7! is closed by using the chargquasineutrality condition, namely,qdnd1eni2ene50, to-gether with a suitable equation of state for the dust comnent. While the present analysis can be carried out for geral equations of state discussed in the literature,43–45we usebelow, to keep the analysis simple, the well-known vanWaals equation of state, namely,
S pd1a
vdm2 D ~vdm2b!5RdTd , ~8!
wherevdm is the molar volume,Td is the dust fluid temperature, anda, b, andRd are the usual gas constants.46 In termsof the dust number densitynd[L/vdm where L is theAvogadro number, Eq.~8! can be rewritten as
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e,hend
ale-nge
-
-n-
r
~pd1And2!~12B nd!5ndkdTd , ~9!
whereA5a/L2, B5b/L, and kd5Rd /L. Note that for anideal dust gas,kd[kB , and Eq.~8! reduces to the usual ideagas law whenA5B50. The constantsA and B are calcu-lated in terms of the critical parameters by requiri]pd /]nd50 and]2pd /]nd
250. This yields the relations
A59
8
kdTc
nc, B5
1
3 nc, ~10!
where the subscript ‘‘c’’ denotes the respective values at thcritical point.
We now consider a steady state equilibrium of tplasma relevant to~parallel! Kelvin–Helmholtz configura-tion, which was first discussed by D’Angelo11 for the usualelectron-ion plasmas, and later extended to dusty plasmAccordingly, in equilibrium, there exists a transverse shin the dust flow velocity component parallel to the ambiemagnetic field.12 Thus, the equilibrium dust flow velocitycan, in general, be represented as
vd05Vyy1Vzz, ~11!
where the componentsVy and Vz are to be suitably pre-scribed consistent with the existence of a zeroth-order stestate as governed by Eqs.~3! and ~4!. The latter are identi-cally satisfied provided the equilibrium quantitiesnd0 , f0 ,pd0 andVz are functions ofx only, while the transverse flowvelocity Vy is a constant given by@cf. x-component of Eq.~4!#,
Vy5c
B0
]f0
]x1
c
nd0qd0B0
]pd0
]x, ~12!
where the right-hand side contains the contributions arisfrom theE3B drift due to the equilibrium electric field andthe diamagnetic drift due to the dust pressure. It maynoted that the special case when the flow velocityVy is zerois also included.
III. DISPERSION RELATION
The above system of equations constitutes a completefor low-frequency electrostatic modes and instabilities inlow-b dusty plasma with transverse shear in the field-alignflow by incorporating the contributions due to the gracharge fluctuations as well as nonideal effects. For weakdients in the equilibrium quantities, the wave spectrum cbe determined by carrying out the normal mode analywhich yields the local dispersion relation. Accordingly, wexpand the various quantities asC5C01C1, where C[(nd ,vd ,qd ,pd), and the subscripts ‘‘0’’ and ‘‘1’’ denote,respectively, the equilibrium and the perturbed quantitiThe latter are assumed to vary as;exp@i(kyy1kzz2vt)#wherek[(ky ,kz) is the wave vector in they-z plane andvis the wave frequency.
Before discussing the full dispersion relation, it maynoted that grain charge fluctuation leads to the existencenovel, purely damped mode16,18which always couples to anyother mode~s! of the system. This is a generic feature
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2352 Phys. Plasmas, Vol. 6, No. 6, June 1999 N. N. Rao
dusty plasmas in contrast to the usual electron-ion plasmand can be readily seen by considering the perturbation etion from Eq.~5!, which yields
vqd152 iv1qd12 iRv2f1 , ~13!
wherev5v2k•vd0 is the Doppler shifted wave frequencand
v15x1ne0Re2S 8p
kBTemeD 1/2
expFe~c01f0!
kBTeG ,
~14!
v25v12ec0
kBTix,
denote the charging frequencies. In Eq.~14!, we have de-noted
x5R
A2p
vpi
lDiexpS 2
ef0
kBTiD , ~15!
where vpi5(4pni0e2/mi)1/2 is the ion plasma frequency
and lDi5(kBTi /4pni0e2)1/2 is the ion Debye length. Thecharging frequenciesv1 andv2 are proportional to the sizeof the grain radius, while for typical parameters for dusplasmas found in the near-Earth environment, they aresimilar order of magnitude.28 Thus, for small grain sizes, thsecond term in Eq.~13! can be neglected, leading to thpurely damped mode,v52 iv1. It follows from Eq. ~13!that this mode exists even when there are no potential fltuations associated with the other modes of the plasma.
To derive the dispersion relation, we linearize Eqs.~3!,~4! and ~9!, and Fourier analyze to obtain the perturbatirelations
nd1
nd05
1
vS k•vd12
i
nd0
]nd0
]xvd1xD , ~16!
2 i vvd11vd1x
]Vz
]xz52
iqd0f1
mdk2
qd1
md
]f0
]xx
1qd0
mdc~vd13B0!1
qd1B0Vy
mdcx
2ipd1
mdnd0k1
nd1
mdnd02
]pd0
]xx, ~17!
pd15F nd0kdTd
~12Bnd0!222And0
2 G nd1
nd0, ~18!
while the quasineutrality condition leads to the relation,
nd1qd05Fne0e2
kBTeexpS ef0
kBTeD1
ni0e2
kBTiexpS 2
ef0
kBTiD
1Rv2nd0
~v12 i v !Gf1 . ~19!
Equations~16!–~19! are combined with Eq.~13! to yield thegeneral dispersion relation
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s,a-
of
c-
v2F v~C6ky1Vd02 S!1Vd0S C4kyS1
C6
nd0
]nd0
]x D G3~v22C5ky2C6kz!
5@~v22kzC6!~Vd0v1kyC4!1C4C6kykz!]
3F vS vVd0S1C6
nd0
]nd0
]x D1Vd0ky~C62C5S!G ,~20!
where S[(]Vz /]x)/Vd0 is the dimensionless parametcharacterizing the shear flow,Vd05qd0B0 /mdc is the dustgyro-frequency, and the coefficientsCi , i 51, . . . ,6 are de-fined by
C15 nd0kdTdF S 12h
3 D 22
29ah
4 G ,C252
Rv2
~v12 i v !, C35
1
4plD2 F11
f v2
~v12 i v !G ,
~21!
C451
mdnd0S ]pd0
]x D ~v12 i v !
~v11 f v22 i v !,
C55ky
mdnd0S C11
qd02 nd0
2
C3D , C65
kz
kyC5 ,
wherea5Tc /Td andh5nd0 /nc characterize the strength othe nonideal contributions. We define the effective Deblength lD by lD
225lDe221lDi
22, where lD j
5(kBTj /4pNj 0e2)1/2 is the electron~ion! Debye length forj 5e ( i ), and the equilibrium number densities are given
Ne05ne0 expS ef0
kBTeD , Ni05ni0 expS 2
ef0
kBTiD . ~22!
In Eqs. ~21!, the parameterf is defined byf [4pnd0RlD2
;RND /lD , whereND is the number of dust grains in thDebye sphere, and is a measure of the dust grain packThe dust component may be considered as tenuous or daccording tof !1 or f *1. For the case when the electroand the ion temperatures are of the same order, the paramf is proportional to the ratio of the plasma potential to tdust surface potential, and is essentially the same as therameterP defined by Havneset al.47 For a tenuous plasma( f !1), the dispersion relation~20! predicts the purely damping modev52 iv1 discussed above.
Before discussing the general dispersion relation~20!,we consider first some limiting cases. For a homogenedusty plasma without any shear flow (S50), Eq. ~20! hastwo solutions for parallel propagation (ky50). First, for thecase when the charge fluctuation effects can be negle( f→0), we obtain
~v2k•vd0!25kz2CD
2 ~11e!, ~23!
wheree[evr1ec f , and
evr5bh~62h!
~32h!2, ec f52
9abh
4, ~24!
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2353Phys. Plasmas, Vol. 6, No. 6, June 1999 N. N. Rao
represent, respectively, the contributions due to the volureduction coefficient and the attractive cohesive forcesEq. ~23!, we have defined,CD
2 /md)1/2 is thedust plasma frequency,Vtd5(kdTd /md)1/2 is the dust ther-mal speed,b5b/(11b), and b5Vtd
2 /CDA2 . Equation~23!
represents the dispersion relation for the usual DAW wnonideal effects.15,35 For the ideal case but with charge flutuations, we obtain
v25kz2CDA
2 2mdRCDA
4 kz2
qd02
v2
~v12 i v !, ~25!
which leads to damping of DAWs due to charge fluctuatioas discussed elsewhere.8,28 For ky50, Eq. ~20! has the sec-ond solution,v25Vd0
2 , which simply represents gyro motioof the dust particles at the Doppler shifted frequency. Onother hand, for almost perpendicular propagation (kz;0) ina homogeneous ideal plasma with no shear flow (S50), Eq.~20! reduces to
v25Vd02 1ky
2CD2 , ~26!
which is the usual normal mode dispersion for the elecstatic dust cyclotron wave~EDCW!.29,30 These modes arealso damped due to charge fluctuations, as discussed be
The general dispersion relation~20! can be rewritten in amore compact form in terms of dimensionless quantitiesCn ,n51,2,3,5, which are defined by
C159
~32h!22
9ah
4, C25
v2
~v12 i v !,
~27!
C3511 f C2 , C551
~11 f C2!1bC1 .
In equilibrium, the pressure gradients give rise to the ddiamagnetic drift defined byVD[ yVD , where VD
5(]pd0 /]x)/mdnd0Vd0. For the equilibrium pressurepd0
given by the van der Waals equation~9!, we obtain
VD5l1bVdn1l2bVdT , ~28!
where the quantitiesl1 and l2 are defined in terms of thenonideality parametersa andh by
l159F 1
~32h!22
ah
4 G , l253
~32h!. ~29!
Note thatl1 ,l2→1 for the ideal case. In Eq.~28!, Vdn andVdT denote, respectively, the characteristic drifts in the prence of density and temperature gradients, and are define
Vdn5CDA
2
LnVd0, VdT5
CDA2
LTVd0, ~30!
where the gradient scale lengthsLn andLT are defined by
Ln5nd0S ]nd0
]x D 21
, LT5TdS ]Td
]x D 21
. ~31!
The general dispersion relation~20! now takes the form
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en
h
,
e
-
w.
st
-by
C3~v22kz2CDA
2 C5!~v22Vd02 !2v2ky
2CDA2 C3C5
5Vd0S v
Ln1kzVd0SD ~vVD1kyCDA
2 C3C5!
1vVd02 kyVD . ~32!
It may be noted that the right-hand side contains contritions arising from the shear flow as well as the drifts duethe number density and temperature gradients.
IV. ELECTROSTATIC MODES
The dispersion relation~32! admits wave modes in thelow-frequency (v!Vd0) regime as well as the dust gyrofrequency (v;Vd0) regime. We consider these two casseparately.
A. Low-frequency regime „v!Vd0…
1. Dust-acoustic wave (DAW)
For a homogeneous dusty plasma for frequenciesv!Vd0, Eq. ~32! yields
v2~11ky2rDA
2 C5!5kz2CDA
2 C52kykzCDA2 SC5 , ~33!
whererDA5CDA /Vd0 is the characteristic dust gyro-radiudefined with respect to the DAW phase speed. For pureallel propagation (ky50), Eq. ~33! generalizes the DAWdispersion relation~23! to include finite Larmor radius ef-fects. On the other hand, the shear flow effects comeplay for oblique propagation when the perpendicular wanumberky is also included as shown by Eq.~33!. In eithercase, the charge fluctuation leads to damping of the mowhile the damping rate can be calculated as discuselsewhere.8,28
2. Density-gradient drift wave (DGDW)
Consider the low-frequency limit of Eq.~32! for a coldinhomogeneous dusty plasma when the wave propagatioalmost perpendicular to the magnetic field but with a smkz so that the electrons and ions can flow along the field lito maintain quasineutrality. Equation~32! reduces to
v52kyVdn
~C31ky2rDA
2 !. ~34!
When the charge fluctuation effects can be neglected, thaC3→1, Eq. ~34! becomes
v52vdn
!
~11ky2rDA
2 ![vdn
! , ~35!
wherevdn! [kyVdn is the diamagnetic drift frequency due t
dust density-gradient. Equation~35! represents a densitygradient driven drift wave6 in a cold plasma. For weak damping, we writev5v r1 ig with ugu!uv r u. Equation~34! thenyields, in the lowest order, the real frequencyv r'vdn
! , whilethe damping rate is given by
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2354 Phys. Plasmas, Vol. 6, No. 6, June 1999 N. N. Rao
g'2f v2v r
2
~11ky2rDA
2 !~v121v r
2!. ~36!
It should be noted that the damping rate vanishes, aspected, in the limit off→0.
3. Temperature-gradient drift wave (TGDW)
Dusty plasmas support a novel kind of drift wave drivpurely by dust temperature gradient even when the dengradient is absent. Like in the previous case, we assummost perpendicular propagation but with a smallkz so thatquasineutrality is maintained by the lighter species. Dispsion relation~32! then reduces to
v52bl2kyVdT
@C31ky2rDA
2 ~11b C1C3!#. ~37!
For the case when charge fluctuations can be neglectedabove equation reduces to
v52bl2vdT
!
@11ky2rDA
2 ~11e!#[vdT
! , ~38!
wherevdT! [kyVdT is the diamagnetic drift frequency due
the dust temperature gradient. The dispersion relation~38! issimilar to ~35!, and represents temperature-gradient drivdrift wave ~TGDW! in a warm dusty plasma. For a tenuoplasma when the damping is weak, Eq.~37! yields real fre-quency asv r'vdT
! , whereas the damping coefficient (g) isgiven by
g'2@11ky
2~brDA2 1erD
2 !#
@11ky2rD
2 ~11e!#
f v2v r2
~v121v r
2!, ~39!
whererD[CD /Vd0. This is similar to the damping rate obtained for the DGDW in the previous case@cf. Eq. ~36!#.
B. Dust gyro-frequency regime „v;Vd0…
We discuss now wave propagation in the dust gyfrequency range, and consider first the case of homogenplasmas. For almost perpendicular propagation (ky
2@kz2)
without shear flows, Eq.~32! becomes
v25Vd02 1ky
2CDA2 ~v12 i v !
~v11 f v22 i v !
1ky2CDA
2 F 9b
~32h!22
9abh
4 G , ~40!
which is a generalized dispersion relation for the electrostdust cyclotron wave~EDCW! modified by the effects due tocharge fluctuations~the third term! as well as nonideal contributions ~the fourth term!. For an ideal plasma with constant dust charge, we recover the dispersion relation~26! forthe electrostatic dust cyclotron wave~EDCW!.29,30 On theother hand, for the nonideal case with weak damping,obtain
v r25Vd0
2 1ky2CD
2 ~11e!, ~41!
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x-
ityal-
r-
the
n
-us
ic
e
and
g52f ky
2CDA2 v2
2~v121v r
2!. ~42!
From the general dispersion relation~32!, it follows thatfor all the modes the effects of shear flows manifest theselves for finitekz . Accordingly, for homogeneous plasmaEq. ~32! reduces to
~v22kz2CDA
2 C5!~v22Vd02 !2v2ky
2CDA2 C5
5kykzVd02 CDA
2 C5S, ~43!
which shows the coupling between the DAW and the dcyclotron mode. To keep the analysis simple, we shall csider the case whenf→0 so that charge fluctuation dampineffects can be neglected, that is,C5→C[(11bC1). Equa-tion ~43! is a biquadratic inv2, and yields the rootsv6
given by
v62 5 1
2@Vd02 ~11k2rDA
2 C!6$Vd04 ~11k2rDA
2 C!2
24kykzCDA2 Vd0
2 C~k2S!%1/2#, ~44!
where k5kz /ky and k25ky21kz
2. Clearly, the modev
5v1 is the usual EDCW modified by the dust shear flowhile v5v2 is a low-frequency mode which becomes ustable for S.k. This can be seen for the case whukykzrDA
2 C(k2S)u!(11k2rDA2 C)2, which leads to
v12 'Vd0
2 1k2CDA2 C1
kykzCDA2 C~S2k!
~11k2rDA2 C!
, ~45!
and
v22 52
kykzCDA2 C~S2k!
11k2rDA2 C
. ~46!
Clearly, as pointed out earlier,v2 is a purely growing, zeroreal frequency unstable mode forS.k.
For inhomogeneous plasmas, the dispersion relation~43!needs to be generalized to include contributions from teperature and density gradients. For the caseuvu@v1, Eq.~32! reduces to the form,
v2@v22$~11A1!Vd02 1mk2CDA
2 %#2vVd02 A2
1mkykzCDA2 Vd0
2 ~k2S!1 i f v2v~v22Vd02 !50, ~47!
where the contributions due to the inhomogeneities athrough the coefficientsA1 andA2 defined by
A15bvdn
!
CDA2 ky
2 ~l1vdn! 1l2vdT
! !,
~48!A25mvdn
! 1b~11kS!~l1vdn! 1l2vdT
! !,
andm5(11e)(11b).Equation~47! is of the formDr1 iD i50, while the pres-
ence of the second term leads to damping of modes giveDr(v,kW )50. For weak damping, we write,v→v r1 ig with
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es-
paon
m
-g
f t
in
geity-w
rge
fy
ar
de-
ed
Ha
it,
ldthehene-
w
f
re-en
2355Phys. Plasmas, Vol. 6, No. 6, June 1999 N. N. Rao
ugu!uv r u, and the damping rate (g) is calculated using theusual expression,g52Di(v r ,k)/(]Dr /]v r), to obtain
g52f v2v r~v r
22Vd02 !
2v r@2v r22$Vd0
2 ~11A1!1mk2CDA2 %#2Vd0
2 A2
. ~49!
For the frequency range, v r2;VDC
2 [Vd02 (11A1)
1mk2CDA2 , we obtain
g52f v2VDC~A1Vd0
2 1mk2CDA2 !
~2VDC3 2Vd0
2 A2!. ~50!
Note that the coefficientA2 is proportional to the drift fre-quenciesvdn
! andvdT! . For the case of weak inhomogeneiti
such thatVDC*Vd0@A2, we derive the simplified expression
g52f v2
2VDC2 ~A1Vd0
2 1mk2CDA2 !, ~51!
which shows the damping of the modes.The effects of the inhomogeneities on the mode pro
gation can be calculated from the dispersion relatiDr(v r ,k)50. For the frequency rangev*Vd0@A2, wedrop for simplicity the linear term inv and obtain the roots
v62 5 1
2@VDC2 6$VDC
4 24mkykzCDA2 Vd0
2 ~k2S!%1/2#. ~52!
As earlier@cf. Eq. ~46!#, the modev5v2 is unstable whenS.k. Furthermore, whenumkykzCDA
2 Vd02 (k2S)u!VDC
4 , weobtain the expressions
v12 'VDC
2 2mkykzCDA
2 Vd02 ~k2S!
@Vd02 ~11A1!1mk2CDA
2 #, ~53!
v22 '2
mkykzCDA2 ~S2k!
~11A11mk2rDA2 !
. ~54!
Equations~53! and ~54! generalize, respectively, Eqs.~45!and~46! to include contributions due to the density and teperature gradients.
V. KELVIN–HELMHOLTZ INSTABILITIES
Dusty plasmas are susceptible to~parallel! K–H type oflow-frequency (v!Vd0) instabilities for oblique propagation (ky ,kzÞ0) in the presence of field-aligned flows havinshear in the transverse direction (SÞ0). We consider belowthe different cases which arise depending on the nature oplasma inhomogeneity.
A. Density-gradient Kelvin–Helmholtz „DGKH…
instability
Consider first the case of a cold dusty plasma havonly density gradient. In the low-frequency limit (v!Vd0),Eq. ~32! reduces to
v2S 11ky
2rDA2
C3D 1
vdn! v
C3
2kykzCDA
2
C3
~k2S!50, ~55!
Downloaded 13 Apr 2013 to 131.211.208.19. This article is copyrighted as indicated in the abstract.
-,
-
he
g
which includes finite Larmor radius as well as the charfluctuation effects. For oblique propagation, the densgradient K–H~DGKH! instability arises when the shear flospeed satisfies the condition
~vdn! !214kykzCDA
2 ~11ky2rDA
2 !~k2S!,0, ~56!
where we have neglected the damping due to the chafluctuation (C3→1), assuming it to be weak.~This will beconsidered separately later.! Thus, for the excitation ofDGKH instability, the shear flow parameter should satiskS.0 such that
S:k11
k
Vdn2
@4CDA2 ~11ky
2rDA2 !#
for k:0. ~57!
Assuming k.0, the instability is excited when the sheflow parameterS exceeds a threshold value (Sc) given by,Sc5Vdn /CDA[CDA /Vd0Ln , where we have assumeky
2rDA2 !1. Thus, ifDVz denotes the relative flow speed b
tween adjacent layers over a scale lengthLn , that is,]Vz /]x;DVz /LN , DGKH instability is excited providedDVz is of the order of or larger than the DAW phase spe(CDA). This is consistent with the result reported earlier.12
B. Temperature-gradient Kelvin–Helmholtz „TGKH…
instability
We now show the existence of a novel type of K–instability which is driven by the temperature gradient inwarm dusty plasma. Accordingly, in the low-frequency limthe general dispersion relation~32! yields
v2~11ky2rDA
2 C5!
1bl2vdT
!
C3
~11kS! v2kykzCDA2 C5 ~k2S!50. ~58!
It may be noted that the dispersion relation~58! is comple-mentary to~55!, which is applicable for the case of a codusty plasma with density inhomogeneity. To estimateshear flow required for instability excitation, we consider tcase when the damping due to charge fluctuation can beglected. Accordingly, Eq.~58! predicts TGKH instability forparameters satisfying the condition
@bl2vdT! ~11kS!#214kykzmCDA
2 ~11ky2rDA
2 m!~k2S!,0.
~59!
Thus, the TGKH instability is excited when the shear floparameter satisfies
S:k11
k
b2l22VdT
2
@4CDA2 m~11ky
2rDA2 m!#
for k:0, ~60!
which is valid forkS!1. The critical shear for the onset othe TGKH instability is given by
Sc5bl2CDA
Am Vd0LT
, ~61!
whereky2rDA
2 m!1 has been assumed. Thus, like in the pvious section, ifDVz denotes the relative flow speed betwe
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y
dt
at
slo
ivple
thfog
roththavt
ints
a
is
ta-ng
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e
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eed.ua-
thetherthe
e
lthe
l
e
2356 Phys. Plasmas, Vol. 6, No. 6, June 1999 N. N. Rao
adjacent layers over a scale lengthLT , that is, ]Vz /]x;DVz /LT , the TGKH instability is excited providedDVz isof the order of or larger thanbCD . Since for typical dustyplasmas,b!1, it follows that the TGKH instability can beexcited even when the relative speed between adjacent lais much smaller than the DAW phase speed (CD) in a warmdusty plasma. Comparing this result with the one obtainethe previous section, we note that the threshold shear forexcitation of TGKH instability is much smaller than threquired for the onset of DGKH instability.
C. General case
We now consider the general case when both the denas well as the temperature gradients are present. For thefrequency regimev!Vd0 ,ky
2CDA2 /vdn
! , the full dispersionrelation ~32! reduces to
v2~11ky2rDA
2 C5!
1vFvdn! C51
b
C3
~11kS! ~l1vdn! 1l2vdT
! !G2kykzCDA
2 C5 ~k2S!50, ~62!
which contains the dispersion relations~55! and~58! as spe-cial cases. We now analyze the dispersion relation~62! forthe onset of K–H instability in two frequency regimes.
1. Case zvz!v1
As pointed out earlier, the dust charge fluctuations grise to a purely damped zero-frequency mode which couto the other plasma waves~cf. coefficientC2) leading eitherto a damping of the normal modes or to a reduction ingrowth rate of the excited instabilities. On the other hand,wave frequencies (v) much smaller than the dust charginfrequency (v1), we find C2→d[v2 /v1. Thus, in this fre-quency regime, the zero-frequency mode is decoupled fthe other modes. Physically, this arises due to the factwhen the grain charging time scale is much smaller thanwave period, the grains have sufficient time to attain anerage equilibrium charge and hence any damping due tocharging processes should be small.
In this limit, C3→(11 f d), and C5→u/(11 f d) whereu[m1(m21) f d. Accordingly, Eq.~62! yields
v2~11 f d1ky2rDA
2 u!
1v@uvdn! 1b ~11kS! ~l1vdn
! 1l2vdT! !#
2kykzCDA2 u ~k2S!50. ~63!
Hence, the condition for the onset of the K–H instabilitythe presence of density as well as temperature gradienobtained as
@vdn! $u1bl1~11kS!%1vdT
! bl2~11kS!#2
14kykzuCDA2 ~11 f d1ky
2rDA2 u!~k2S!,0. ~64!
For kS!1, the instability is excited for shear flows such th
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ers
inhe
ityw-
es
er
mate-
he
is
t
S:k11
k
@~u1bl1!Vdn1bl2VdT#2
@4uCDA2 ~11 f d1ky
2rDA2 u!#
for k:0. ~65!
The critical shear for the onset of the K–H instabilityobtained as
Sc5CDA
Vd0F ~u1bl1!
Ln1
bl2
LTG 1
Au~11 f d!, ~66!
whereky2rDA
2 u!1 has been assumed.The effect of dust thermal pressure on the DGKH ins
bility discussed in Sec. V A can now be calculated by takithe limit LT→` in Eq. ~66!, which yields
Sc5CDA
Vd0Ln
~u1bl1!
Au~11 f d!. ~67!
The corresponding critical relative speed necessary to dthe instability is found to be
~DVz!c'CDA
~u1bl1!
Au~11 f d!. ~68!
For an ideal, warm (b!1) dusty plasma, we obtain in thlowest order
~DVz!c'CDA
~11 32b!
A~11 f d!. ~69!
The critical speed between adjacent layers necessary foonset of the DGKH instability increases due to dust thermcontribution, while denser plasmas have lesser critical spFurthermore, though wave damping due to charge flucttions has been neglected, there is a overall decrease incritical speed because of the charging process. On the ohand, when the nonideal effects are included, we obtainexpression
~DVz!c'CDA
~m1bl1!
Am~11 f d!. ~70!
For weakly nonideal dusty fluid, this reduces to
~DVz!c'CDA
~11 12e1 3
2b!
A~11 f d!. ~71!
It should be pointed out that the contribution (evr) due to thevolume reduction factor is typically positive while that duto the cohesive attractive forces (ec f) is negative.15 Accord-ingly, the net contribution (e5evr1ec f) due to the nonideaeffects can either be positive or negative depending onrelative values of the nonideality parametersa and h. Fora!1, one finds15 thate.0 for sub- as well as super-criticavalues ofh. On the other hand, larger values ofa lead tonegative contributions frome. Thus, the critical relative flowspeed (DVz)c for the onset of the instability depends on thspecific values ofe andb.
On the other hand, for the TGKH instability, Eq.~66!reduces to
Sc5l2bCDA
Vd0LT
1
Au~11 f d!. ~72!
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th
abd, w
-thin
le
tuhela
thectcla
-
, as
nly
q.ak
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rytedseare
dout
t in
r.
ef-nsesita-e-
areustte.
de-ionof
w-ithua-ra-styro-ronch
2357Phys. Plasmas, Vol. 6, No. 6, June 1999 N. N. Rao
The corresponding critical relative speed for the onset ofTGKH instability is given by
~DVz!c'l2bCDA
Au~11 f d!. ~73!
For the ideal, warm (b!1) case, this reduces to
~DVz!c'CD
b
A~11 f d!. ~74!
Hence, it follows that, like in the previous case, the criticrelative speed necessary for the onset of the TGKH instaity decreases for denser plasmas as well as due to thecharging process. When the nonideal effects are includedobtain the expression
~DVz!c'l2bCD
A~11e!~11 f d!. ~75!
For the weakly, nonideal case, Eq.~75! yields, in the lowestorder,
~DVz!c'l2bCDA
~12 12e2 1
2b!
A~11 f d!. ~76!
Comparing this expression with~71!, we note that the contributions due to the nonideal as well as thermal effects toDGKH and TGKH instabilities are quantitatively oppositenature.
Finally, we mention that when the two gradient scalengths are of the same order, we get the expression
~DVz!c'CDA
~11 12e1 5
2b!
A~11 f d!. ~77!
In the above discussion, we have ignored the charge fluction damping, and hence the K–H instability would have tgrowth rate as predicted, in general, by the dispersion rtion ~63!. In the next section, we include the charge fluctution to estimate the wave damping.
2. Case zvz@v1
We now consider the casev@v1 when the grains do nohave sufficient time to attain an equilibrium value during twave period. Hence, the plasma modes would see a fluating charge, and would be coupled to the zero-frequenpurely damped mode discussed earlier. The effect of theter manifests itself through the coefficientC2 which, for thepresent case, becomesC2→ i v2 v while, in the lowest or-der, C3→(v1 i f v2)/v and C5→m/C3 for a weakly non-ideal, tenuous dusty plasma. Thus, forf Þ0, one expects areduction in the growth rate of the instability.
For the present case, the dispersion relation~62! reducesto
v2~11ky2rDA
2 m!1v@mvdn! 1b ~11kS! ~l1vdn
! 1l2vdT! !#
2kykzCDA2 m ~k2S!1 i f v2v50, ~78!
Downloaded 13 Apr 2013 to 131.211.208.19. This article is copyrighted as indicated in the abstract.
e
lil-uste
e
a-ea--
u-y,t-
which is of the formDr1 iD i50, and is similar to the dis-persion relation~63! discussed in the previous case. However, Eq. ~78! has an additional explicit imaginary termwhich is nonzero, and gives rise to damping effect. In factexpected, Eq.~78! exactly reduces to Eq.~63! in the limitf d→0 since charge fluctuation effects are appreciable owhen the productf d is finite.
The contribution arising due to the second term in E~78! can be calculated as in Sec. IV B. Accordingly, for wedamping, we obtain
g52f v2v r
2v r~11mky2rDA
2 !1A2
, ~79!
whereA2 is given by Eq.~48!. The conditions necessary fothe excitation of the K–H instability are governed by thdispersion relation,Dr(v,k)[0, which is structurally simi-lar to Eq.~63!. The analysis to obtain the condition necessafor the onset of the instabilities runs parallel to that presenin the previous section. In particular, for the general cawhen both the density and the temperature gradientspresent, the K–H instability occurs provided
@vdn! $m1bl1~11kS!%1vdT
! bl2~11kS!#2
14kykzmCDA2 ~11ky
2rDA2 m!~k2S!,0, ~80!
which is similar to Eq.~64! obtained for the case whenuvu!v1. The corresponding critical shear is given by
Sc5CDA
Vd0F ~m1bl1!
Ln1
bl2
LTG 1
Am~11ky2rDA
2 m!. ~81!
The remaining details for the special cases of Eq.~81! appli-cable for the DGKH and TGKH instabilities can be workeout as in the previous case, and hence will not be carriedhere. In this connection, it may, however, be noted thaview of the similarities between Eqs.~66! and~81!, the spe-cific details in the two cases would be qualitatively simila
VI. SUMMARY
To summarize, we have investigated the combinedfects of nonideal contributions and grain charge fluctuatioon the propagation of low-frequency electrostatic modsuch as the DAWs and drift waves as well as on the exction of gradient-driven K–H instabilities in a dusty magntoplasma. For these modes, the lighter electrons and ionsassumed to behave like ideal fluids while the heavier dfluid is described by the van der Waals equation of staStarting from the relevant governing equations, we firstrive a generalized dispersion relation for oblique propagatwith respect to the ambient magnetic field. Different casesthe dispersion relation are then analyzed. In the lofrequency regime, DAW and drift waves are recovered wmodifications due to the nonideal as well as charge flucttion contributions. Our results show a novel kind of tempeture gradient-driven drift wave which exists even in a duplasma without density inhomogeneity. In the dust gyfrequency regime, we obtain the electrostatic dust cyclotmode which is modified by the shear flow. The other bran
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floobea
tylityyethatethr
ntic
uctio
m
hys.
ett.
J.
. E
ys.
.
2358 Phys. Plasmas, Vol. 6, No. 6, June 1999 N. N. Rao
of the cyclotron mode becomes unstable when the shearexceeds a threshold value which is determined by the ratithe parallel to perpendicular component of the wave numIn general, the charge fluctuation leads to an overall decrein the growth rate of the excited instability.
For the K–H configuration, we obtain an instabiliwhich is driven by the temperature gradient. This instabiexists when the relative flow speed between adjacent laexceeds a critical value which is much less than that forcase of the density gradient-driven K–H instability. The lter typically occurs for flow speeds of the order of or largthan the DAW phase speed. It is found that in either case,critical flow speed reduces because of the charging pcesses. On the other hand, the effect of the nonideal cobutions in the two cases shows a quantitative behavior whis complementary to each other. For frequencies msmaller than the charging frequency, there is a net reducin the growth rate of the instabilities.
ACKNOWLEDGMENT
This work was partially supported by the Abdus SalaInternational Center for Theoretical Physics~ASICTP!, Tri-este~Italy! under its Associate Membership Program.
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