J. Plasma Fusion Res. SERIES, V4 (2001) 540_s44

Structures in Nonuniform Streaming
Dusty Magnetoplasma

Potential GohI

VRANJES Jovoo and SHUKLA Padma Kant'of Physics, P.O. Box 57, Yu-L1001 Belgrade, Yugoslavia
Physik IV. Ruhr-I|niversit(it Bochum, D-44780 Bochum, Germany

Institut fiir

5 December 2000 / Accepted: 16 August 2001)

Abstract
In this paper wenonuniform dustv
and in basically twov linear and nonlinear behaviour of modified convective cells and vortices inwith perpendicular and parallel to the magnetic field plasma flows,physical systems, with stationary (corresponding to the case of Shukla-Varma mode) anddust, by choosing someeigenvalue equation in
separating spatially

ific profiles for the sheared plasma flow and the dust density, we analyze theto deduce the growth rate. A threshold is also obtained for the wavenumber
and convective modes (growing in space) due to its interaction with theary (i.e. taking part in perturbations) dust particles. For the case of stationary

nonlinear regime, for both stationary and nonstationary dust particles, and inflows perpendicular and parallel to the magnetic field lines, a variety ofthe presence of various
possible nonlinear solu
driven by the nonuniform shear flow and dust density is presented, i.e.,single and double vorte. chains accompanied with zonal flows, and tripolar and global vortices.
sheared plasma flow. In

Keywords:
shear flow. coherent, vortex chain, tripole

1. Stationary Dust. TheDerivations
We use the model [1]is) nonuniform flow

From the ion momentum equation, the ion andelectron continuity equations, and by using Poisson'sequation, we obtain, tor ldl)tl ( C)1 = eBnlmi, thefollowing expression for the conservation of the charge

is thendensity:a nonuniform, three-component electron-ion-duy plasma with anequilibrium perpendicular (wmagnetic field Bs?, along the(drift) % = vo@)Vr. The basic

respect to a uniform

v,@y=unperturbed number of charges residing on the surface
ofthe dust grains which are negatively charged.

andjr4eo@f (r) = efnn(x)- ny1
where @6 is the unperturbed

a2@oldf, nis is the unparticle speciesj (j equals eions. and d for the dust*. i;.,x va(o + @o) vr.

)+Zun*(x)l, (l)Ia1v]o+ a'i@)-of (x)+ fi(na- n,ll= o, Q)tpotential, O'i =number density of thethe electrons, i for the-"lor, for the total potential Y = @ + @o asains), and Za is the02001 by The Japan Society of Plasma*Corresponding author's e-Science and Nuclear Fusion Research: vranj e s @ phy. b g.ac.yu
540

Vranjed J. et al., Potential Coherent Structures in Nonuniform Streaming Dusty Magnetoplasma

xv v.v,lIdt * -L;Bo"II-rForz6s satisfying nao= Botodalog [1 - tanh2(tcx)]l2eZo,

the solutions of Eq. (5) become Legendre functions ofthe degree 1, and ofthe order y= (kj + l)tt2lrc. The only

t.#,. (3)localized solutions for 4 --> +1, i.e. for-r -+ 16, areobtained if v2: (G + D/r2 = 1. The wave becomes

Here the prime denotes derivative with respect to the x-

convectively unstable (growing in space) because oftheresonant interaction with the flow, provided that thewavenumber satisfies the condition kr> (rc2 - 1)1/2. In["vlv- *"*uri,=o,0,=coordinate. The nonlinear equation (2), which describes

the evolution of the electrostatic potential in a dusty

that case, assuming a small deviation 6'ft, of the stable

plasma containing a nonuniform background of
stationary dust grains as well as equilibrium density

value of the wavenumber ftr, and taking ar -) at + i@i,
one can find the following expression for theinhomogeneities and sheared plasma flows, is analyzed
both in the linear as well as in nonlinear regimes.

corresponding growth rate
dvo@._46ky3z-I)l-'-rcn(z*-DI d, l,="2. Linear Analysis
Within the framework of a local analysis, we obtainIn the opposite limit, the wave is damped.
Now, for the given profiles of v6(.t) and na6(.r),

from Eq. (2) the dispersion equation for the ShuklaVarma (SV) mode [2] that is modified due ro the shear
plasma flow. In the nonlocal case, for the perturbations

using the Rayleigh condition we find the following
connection between the flow width rl and the criticalpoints for the instability tanh(r"r")= (l - ll(2K2))1t2.
Thus, an absolute instability at the point x. for modes
with wavenumbers satisfying this condition can beexpected. Furthermore, it follows that there appears aof the form <D = 6 exp(-iat + ikyl), the necessary
condition (viz. the Rayleigh type) for the instability canbe readily found by assuming a complex frequency ro =at + ia)i, and integrating perpendicular to the flow
direction, i.e. in the x direction. We have the Rayleighlimiting value Kl = 0.71 above which the critical point x"
becomes complex, i.e., the Rayleigh type instabilityconditionv,!(i_ ezon'*(x) =0.
Bo€oa(6)vanishes.

(4)Thus, an absolute instability is possible if at any point x.
in the transverse direction (perpendicular to the directionof the flow), the condition (4) is satisfied. The presence

3. Nonlinear Solutions
For a linear dust density profile nae(.r) = uByesqxl
eZ6, nonlinear Eq. (3) can be integrated. Further, wesearch for solutions of the form Y(r, y) = Yr(-r) +Y2(x) cos(ftry), where lYr(x)l > lYz(*)|, which yields theof the dust density gradient modifies the standard
condition requiring the existence of an inflection pointin the flow profile.

following set of equations

Further linear stability will be discussed moreclosely for some specific profiles of the shear flow andthe dust number density. We suppose that the flow
profile is described by the following well behaved

ltc
IO-

\dx't)v,r'r"

2coshl V,lrSuno*1' A)I

analytical function v6(x) = u + aKt^nh(Kx), where r-r

f^+-ri-rjv,r"rdefines the characteristic width (slope) of the shear flow.

\dr'We now proceed calculating appropriate derivatives ofvs(.r) and introducing a new variable 4 = tanh(Kr), andby assuming a physically interesting case when u = o)lftr. The linear equation for the perturbed potential can bewritten as:=-BY2@)

sinhIY, (x)-uBox]
2cosh21Y, (x)-uBoxl(8)Equations (7) and (8) can be solved numerically. Well(t -n2) t9 -2n4+[r-1:l -tdq' dq I
Kz_l-n'.#in.d#*ryld'=ol

localized (in the.r-direction) solutions in the form of adouble vortex chain associated with a strong gradient of
the potential in the region between two vortex chains,
can be found by solving the above system of equations
from the point x = 0. The strong gradient of the potential(s)Y in the central region, around the point x = 0,I541

Coherent Structures in Nonuniform Streaming Dusty Magnetoplasma

Yranlei L et al.,and strong zonal flowcorresponds to a spatiallyin the direction perpendicu r to the basic stategradients. Consequently, aPartflinks of the two chains fflow, the plasma particles are
move in the x direction bY
experimental situation, suchbeing trapped in thelaterally to the zonal
itionally prevented tozonal flow. In a real>'0

configuration shouldrepresent an efficient barrier or the plasma particle
transport in the -r-direction, i.e in the direction of theplasma nonuniformities.
, it can be shown thatparameters B and k,
there exists a class of values
solutions are possible.
for which the above descriMore details regarding this nu rical procedure can be.3-40x

found in Ref. [3].

of non-propagating,
A completely different ti.e., carried by the flow, non inear solutions can be
I <D starting from Eq.
found for the perturbed
I <Do(-r), i.e., for(2), in the case of a quadratic
profile* us, and2aoxlBo
flowlinearshearthe)=for a quadratic dust number
2asendl eZ6- bsenarzl eZ6,parameters that describe theIntegrated Eq. (2) in that cases(Q + asxz), wherc 9 is an
given argument, which we takef'6 + F,. (<D + a6x2). allowing
have different values inside

circle with the radius 16.
solutions independently outs
circle and match the soluthe circle. The quadrupo

Fig. 1 Tripolar vortex for the perturbed potential @,driven and carried by the plasma flow. Lateral
vortices have opposite direction of rotation withrespect to the vortex core. The step between
contours is 1.

ity distribution ra6(x) as, bs ara physicalfrom the assumption that the given circle rs is an isolineof the argument of 9. These conditions comprisecontinuity of the potential O and the Laplacian V2O at
the given border. Although the analytical expressions(9), (10) consist of monopolar and quadrupolar parts, inin the basic state.
Y21Q_ + bsxz =trary function of thethe linear form as 9=or the constants Fs.1 tofact they represent a tripolar vortex generated in andcarried by the linear profile of the shear flow, and for aoutside of an arbitraryus, we search for theand inside of the given

quadratic dust number density (Fig. 1).

obtained in that way at

4. Dust-Modified Drift Waves

vortex can be foundIn the previous analysis the dust particles are taken
as stationary, i.e., non-taking part in the perturbed

straightforwardly in the form@ou,(r,0) = do Ko(hor) + dz(e)r > ro,(D,"1r,e;=-t(u'::\ttrft

motion of the electron-ion fluid. In some limits
however, i.e., for very low-frequency processes theirmotion should be taken into account [4]. Assuming,(),6r)cosQ0),

Boltzmann distribution for electrons and using plasma
quasineutrality condition, and continuity and momentum
equations for ions and dust particles in a cylindrical) bn+tan" f- )"
2fr

r_plasma system with perpendicular flow, we obtain:+ coJo(Lr)*frrt,tlO-h ?,,f*"o,,,f ) fo.

(10)

Here, Fsout = 0 and we haveFf"' = bolao tr,
modified Bessel andlv. The constants ofFi" = -h',and Kn.2, Js.2 are-lr, +rr)(@

Bessel functions, respecti
the parametef 16 are tointegration do.z, co,z. Fd", Fi".
continuity conditions at
be found from the atinuity of the function
the given circle, i.e., from the9, the continuity of the gradie of the potential @, and+ oo)- @.(")-

f,(')] = o, (l t)Here we have defined €'o@) = -dllBs4nisln"s' p2 = T"lmit\?, ao = nisln.s * malZ6mi, R is the radius of theplasma column, V6 is some characteristic shear flow542

Vranjei J. et al., Potential Coherenl Structures in Nonuniform Streaming Dusty Magnetoplasma

velocity, and we use the normalization:

A:R g.
atdt

5. Dust Drift Waves
In this section we analyse very low-frequency

V,=RV, .

perturbations in dusty plasma with a nonuniform plasmaVo(@, @0,5r;=(o'3ii')

flow 7s = vs(x)?, which is almost parallel to the lines ofa sheared magnetic field B'e = Bs?. + f(x)Bo?r,lf(x)l <1, and with the dust density gradient in x-direction,.

Bo RVoApart from possible solutions similar to those found inassuming Boltzmann distribution for electrons and ions,the preceding text, steady state solutions of Eq. (1 l) canand using quasineutrality condition. Parallel momentum

be found after integration which yields the following

for dust particles can be written asI'equation:(V1+L')v= e (r),|,xY,e.v,l[r,, + un(x)]l+.+ld, Bo '(r2). . Ev-, eZ" dQ

where k is a constant of integration, and g(r) = -R2((Do+ vo(x)+ €dlaop2.For a quadratic function g, after someEq. (1a) can be integrated yielding vzt = p. @, provided

algebra the solution ofEq. (12) can be written as

Y (r,01that=4 r, t, (kr)cos(n 0) +coJo(kr)+ cr2 + const.,(r4)a;=-ra a,vo (r) +fr(x)= F' I p(x)- Bourxl,fi@)= Ao"f(x) , p'(x) = Bovo @)f (x).

(13)

where cs and c are some constants. Keeping the third
harmonic in the summation only, we obtain the contour
plot of the vortex that is presented in Fig. 2. The(15)

Using Eqs. (14), (15), the dust continuity equation canbe written in the form

structure resembles experimental result which isobtained in a DC discharge in an unmagnetized dusty[#.;uplasma [5]. In the latter, the dust vortex flow is

x v'(o+o,", o']t(l - p?vl)o- Y(r)- 6(")l = o .

generated by a small biased metal plate.

(16)

Here p6 = Ca/{)a, Cza = Z2aT"r/md, Ter = Z26n6sTiT.l
fma(nisT.+ n"sT1)1, Y(-r) = Bs{r^filogn*o, (@) = cf2(x),c = maTerFlez, and, F is a constant of integration.
Tripolar vortex solution of Eq. (16), similar to Eqs.

(9), (10), and traveling with the velocity u, in ydirection, can be found for specific profiles of the basic
state functions/(-r), vs(x), z6(-r) given by the following@equationsu'f(x\= bx(x)+ (x) .
Bo vovo(ou*, , )- Fbx=g.
v,^(x\+2^
yo(x)\4Jt1rrogno(x)l'=EftO^ #,lZ*,,1;fi ftzl

For some profiles of the basic state functions a single
chain type solution, resembling Kelvin-Stuart cat's eyes,
can be written in the formiD(x, Y) = u, Box|- p(x),,\rD
Icos(ft"y)1. (18)-*l\ P't+A log2lcosh(k,r)*lrFig. 2 Global vortex representing a cylindrical chain ofcounter rotating steady state vortices for the totalt

potential in a cylindrical dusty plasma. The steplA solution in the form of a double vortex chain can bebetween contours is 0.5.
543

Coherent Structures in Nonuniform Streaming Dusty Magnetoplasma

Yranjel J. et al.,References

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Bo vo

flog no(r)]'+-I9L

Bort

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ar vortices are novel
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