high frequency drift instabilities in a dusty plasma

8/13/2019 High Frequency Drift Instabilities in a Dusty Plasma

1/6

Pergamon P/tr/lc,r. S/,rrc,c,Qi.. Vol. 41. No. IO. pp. St(L)XYl. I)93Copyright ( 199-l Elsevier Scicncc LtdPrinted ii, Great Britain. All rights roservcd0031~ 0633 Y3 $7.00 +o.oo0032-0633(94)0015.%7

High frequency drift instabilities in a dusty plasmaM. Rosenberg and N. A. Krall Dept. of Electrical and Computer Engineering, University ofCalifornia. Sun Diego. La Jolla. CA Y?OY3. U.S.A Krall Associates. Del Mar. CA 92013. U.S.A.Receiccd 28 t\pril 1994 : revised ?-Z June lYY4 : accepted 1-l June I W-l

Abstract. High frequency drift instabilities witho,, >> C J >> O,, are investigated in a dusty magnetizedplasma in which locally there is an electron densitygradient which is opposite in sign to a dust densitygradient. Two different equilibria are considered. char-acterized by pd 5 f.,,,, where pd is the dust gyroradiusand Lnd is the dust density scale length. Possible appli-cation to Saturns F-ring is discussed.

1. Introduction

Electrostatic instabilities driven by currents associatedwith gradients in the plasma density in a magnetized clcc-tron ion plasma ha\,e been the subject of many invcs-tigations (e.g. Krall, 1968 ; Mikhailovskii, 1974. 19Y3 andreferences therein). Recently. the universal instability hasbeen found to persist also in a multicomponent negativeion plasma. in which locally the electron density gradientis equal to but opposite in sign to the negative ion densitygradient. so that there is no net density gradient or current(Krall. lYY3). In various space environments such as plan-etary ring systems there is thought to exist a plasma inwhich solid particulates. or dust grains, carry a signilicantfraction of the negative charge (c.g. GoertL. 19X9) andwhet-c gradients in the optical depth (which is pro-portional to the dust density) have been reported (c.g.Lane ct (I/.. 19X2). In the case that the spacing betweenthe charged dust grains is much less than the plasmaDebvt: length. the dust grains may be treated as heavy.mult*iply charged negati\,e ions in the plasma ( FortoL andlakubov. IYYO). However. since the mass of a 1 micronsiLed grain is of the order of IO proton masses (assuminga mass density of the order of I g cm--). and its charge-Z,o may be of the order of IO elementary electroncharges in a IO eV oxygen plasma. the charge to massratio of a dust grain in a space plasma is generally much

smaller than that of negative ions. Because of this. themagnitudes of the dust plasma frequency and the dustgyrofrequency IJ~> PJ,, in a dusty magnetired plasma inwhich locally there is an electron density gradient whichis opposite in sign to a dust density gradient. We considertwo equilibrium situations characterized by the relativemagnitudes of the dust gyroradius /I> L ,,,,.Section : gives the equilibria for these t\vo situations. andthe dispersion relation for high frequency drift instabilityfor the two cases is given in section 3. Section 4 presentsthe stability analysis and gives the conditions for insta-bility. The discussion in section 5 considers possible rel-evance of the results to the edges of ringlets in Saturns F-rinrrc.

2. EquilibriaAs shown schematically in Fig. I. a localixd region 01negatively charged dust grains of uniform mass /?I,, andcharge state Z,, (and therefore uniform grain si/e, since1~1,~ 1. and generally Z, x (I) creates an electron hole,with the dust density gradient in the opposite direction tothe electron density gradient at the boundaries. The ion


2/6

x90

density is assumed uniform, as are the temperatures of hethree species (the plasm:i is not assumed to be isothermal.however). WC assume that the spacing hctwecn the dustgrains rf Y liCi L,,,,In this case the clcctrons and ions arc n~~t~netic~tlly con-tined since I)~. it, < L,+ but the dust grains are rlectro-statically confined. so that there is a zero order electricfield in the equilibrium configuration as shown in Fig. 2b.The fluid description at the interface .Y = _Y,or .Y = .Y? s

II,C I ,, B drl,- - T, -- +t7,cE,,C d.\- (7)

(8)

Here E,, is the mvgnitude of the s-component of E . Theclcctron fluid velocity C,,. has contributions from both theclcctron di~~in~l~~ie~ic drift Y[,~= - (r,T,,r7,.c~%)(drl,,d.\-)vand the E x B drift vr = (.E,,c~Rfy, while the uniform ionshave ;I homogeneous E x B fluid drift. Using equation ( I ).the ratio of the electron diamagnetic drift speed to theEx B speed is

rn, Z u T,fE II, T CO (I,, for the configurations of both case (a) and


3/6

M. Rosenberg ancl N. A. Krall : High frequency drift instabilities in ;Ldusty plasma 891

- t l i\ e

A d

i e

Dd

- i TTDe VE\ *e

Dd/ d

/B(x)

2Fig. 2. Slab model geometry used for local analysis. (a) p, < L,,,,(b) Cd Lx,

case (b). We consider a local analysis for the con-figurations shown in Fig. 2. Electrostatic perturbationsare considered with E = -V$, with ti(r,f) = $(.\-)exp[i(X-l,+h-,--(?,t)].

The constants of tnotion for the electrons and negativelycharged dust grains in the configuration of Fig. 2a are thenormalized particle energy H = 13 nd the coordinates ofthe guiding center 73,= s- lliUJ,, (wcy = Z,~B/m,c is thegyrofrequency of species cz) (e.g. see Miyamoto, 1989).Thus for the electrons we take

and for the dust grains we have

The ions are described by a Maxwellian distribution

(14)With these distributions. the electron and dust density

gradients are given by

1 drt,,P -----J - TV,, .\-

(15)

(16)and the magnetic field gradient is given by Amperess law,V x B = (471;~) J, where the cross field equilibrium currentdensity is

Jr = -CA

which leads again to equation (5).Using a standard Vlasov analysis for each species. we

have for the first order perturbed distributions

wheref, = &.Y)cxp [i(k,y+k,:-cd) dt]ul

withdr- = vdt

dv v x B-- = 4 --~--dt 6with r = r v = v. at I = f for each species. Perfotmingthe orbit integrations in the standard way (e.g. Krall andTrivetpiece. 1973) we have for the strongly magnetizedelectrons (0 CC >,,

while for the umnagnetized dust grains with CO>z (Q

Here the electron. dust diamagnetic drift frequencies aregiven by (I>,*= li, inc = I?, T,e,Irl,tucC. t0$ = k, I,)> (I),,


4/6

X9 M. Rosenberg and N. A. Krdl : High frequency drift instabilities in a dusty plasma

.t;, = - y.t;,,+ lY/)T ( -ALI (r)-k*\The dispersion relation is obtained by using

li26) = (/if +/ii) (/I = c 47rq,f;,dr

(21)

to yieldf i ' +h~~c[l +(l - ~~) > (I),,the dust grains have no net fluid drift (SW e.g. Davidsonand Gladd. 1975. for electron-ion plasma). Using thenormalized energy H,, = 1.~ ?~~Z,,&.\-/W,, as the constantof motion for the dust. where I?,, = (T,,,c,Z,,rll,,)(drl,,,d.\-).we take for the dust distribution

(5)while the dust density gradient is given bq

d/l I. the dispersion relation(13) becomes


5/6

M. Rosenberg and N. A. Krall : High frequency drift instabilities in a dust plasma xc)3

(34)where Tll(h,) = I,,(h,) exp ( -h,). Since

using equation ( I ), this becomes

(35)This has the approximate solution

Instability requires finite wavelength. that is finite kE.fj,and (1 > (r+,, which implies a suffciently sharp electrondensity gradient such that

(37)where (5 = 7z,;/7,> I for a dusty plasma with negativelycharged grains. In addition, the assumption > Irequires that

(38)

4.2. CrrscJ (h)Since both the electrons and ions have an Ex B drift here.we do not consider the lower-hybrid drift limit (e.g. Kralland Liewer, 1971) but rather we consider the universalinstability limit (e.g. Krall, 1968) > I. and


6/6

xc)3 M. Rosenberg and N. ,4. Krall : High frequency drift instabilities in ;t dusty plasmau - I ,~m. and for high optical depth regions with r - 0. I,II, - II, (e.g. Goertz, 1989). We consider rather the edgesof rhc ringlets where the optical depth in some cases goesfrom finite values to near zero and assume ZLIqj/q < 1From equation ( I ). the dust density scale lengthL,,, = I( I ,I?~,(dn,,:dr)/- is related to the electron densit ,scale Icngth L,,, = I( I:/I,)(drl,d.v)l by

In the following we assume Z,,U,,:~~, _ I /2 and L,,,, - I kmso that L,,, - 2 km. The dust gyroradius ilLI is related tothe ion gyroradius via /)\, = pl(q,7,~~qTj) I-Z,,. where1?1,,~1?7: - 5 x IO for micron sized grains and oxygen ions.and Z,, - IO for a negatively charged isolated+ grain in a10 eV oxygen plasma (considering only plasma collectioncurrents). We have not included in the analysis the stream-ing ofdust grains across the planets co-rotating magnetictield, which arises because the electrons and ions co-rotatewith the magnetic field. while the grains move with nearlyKeplerian speed (e.g. Hill and Mendis. 1983). For the F-ring, WC estimate this relative stream speed to be of theorder of P&J. which is lower than the phase velocities ofthe modes we are considering which have cqh- >> I,.

Case (a) considered above, corresponding to L)~, L,,,,,would bc applicable only if the dust temperature werelow. with 7;, - (1.01 cV. This is consistent with estimatesof the r;mdom vctocity of grans in planetary rings whichhave been made by balancing the energy gain rate due toviscous stresses with the energy loss rate due to inelasticcollisions giving ;I value of the order of I,, - IO- 111 s-for opticrtliiy thick disks (Border&s PI trf., 19X4; Goertz.19Xc)). Using 1.3 = T,I;~q, gives T,, - 0.01 eV for a I micronsized grain. Since however (II~,:J~I,)~ - 0.0058 for oxygenions. the instability condition (37) would not be satisfiedwith for example /I~ - IO m and iinite kk,,,.

A more likely scenario for the rings ib that the dusttemperature is higher. with Tci 5 T,, since the ion dustthern~~~li~~~ti~~l~itne rJ - f: x 10 t77, Z~/I~~H?, T: 1 is of the order of 30 years using the above parameters;this is much shorter than estimates of ring lifetimes rang-ing from > 4 x I Ohyears to the age of the Saturnian system- 5 x IO years (c.g. Border& ct trl.. 1984 : Northrop and

Connerney. 1986; Alfven. 1954). Then the dust gyroradiusis much larger than L,,,. which corresponds to the caseconsidered in (b). Using ijc - IO 1-11. he instability con-dition (47) would at best be marginally satisfied ; but notethat it is also required for instability that the plasma benonisothermal, with T, > T, 5 - 3:2). It thus appearsthat the ringlet edge would be stable to the excitation ofthis high frequency mode, which is not ilic~~nsistel~t withobservation of the existence of such edges..~~,X/ro~~,/~,cl~/~,/~~~,~~/.s.ne of the authors (M.R.) thanks D. A.

Mendis for holptul dixussions. The authors ~vish to thank DrF. Melandw and another referee for useiul comments. Supportf~rom the following grsnts are gratefully acknowledged b> oneof the authors (:M.R.I : NSF AST-9?1?N36. NASA Origins ofSolar Systems Program NAGW-7152, and Los Aiamox NationalLaborator)lCiPP Grant Y?-137.

References

Borderies. N., Goldreich, P. and Tremaine, S., Unsolved prob-lems in planetary ring dynamics. in P/LIIIc,~~~, R&j,5 (editedby R. Grernberg and A. Brahic). p. 713. Untversity of Ari-ron;t Press. Tucson. 1984.

DAngelo, N., Lox\-frequency elcctrnstatic \v:t\es in dusty plas-tn3s. Phwi. spiccc s1.i. 38, I 143-l 146. 1990.

Davidson, R. C. and Gladd. N. T., A~KI~II~~~LI~ transport propcr-ties associated \vith the lower-hybrid-drift instability. P/tjx./%itl.r 18, 1377. 1975.

Fortov, V. E. and lakubov, I. T., P/~.wic~.\of : o/~i&rl P/rr.w~r,.Cl-rap. 8. Hemisphere. New kork. IYYO.

Fried, B. D. and Conte, S. D. Tir,, ~~f~.s~?~~~~.s/~~,~.s;~~~~~~~~~,~j(}/z.Academic Prcas. NW York. 1961.Goertz, C. K., Dusty plasmas in the solar system. Rust-.C;.w$~~~s.

27, 17 I -2Y7 IYXY..Hill, .J. R. and Mendis, D. A., On the dust current of Saturns F-ring. (;q~//.\~s. Ncs. I~tf. 9, 1069-l 07 I. I YX2.Krall. N. A., Drift wxves. in .~tltw~~cw irr Plrwtrct Pl~~~.sic:r. ol. I

(edited by A. Simon and W. B. Thornp~~~~i). Interscience.New York. 1Y6S.

Krall. N. A., IJnivcrsal instabiliiy in an electron hole tilled withnegative ions. P/IJ.s. F/t/itlr BS. 3 IX-3 13. lYY3.

Krall, N. A. and Liewer, P. C., Lo\v frequency instabilities inmagnetic pulses. Plrr.r. Rcr.. .-I 4, W&C? 10.3. I Y7 I

Krall, N. A. and Trivelpiere, A.. ~~;~~(, j~/~,.~ f fftrww Pi?wi~s.~~~~i-~~\~-~ili, Ncu York. lY73.

Lane, A. L., Hord, C. W., West, R. A., Esposito, L. W., Coffen.D. L., Sato, M., Simmons, K. E., Pomphrey, R. B. and Morris,R. B., Photopolurimctr~ from Voyager ? : preliminary resultson Saturn, Titan, and the Rings. SC%WW215, 537 543. 1981.

Mendis, D. A., Hill, J. R., Ip, W.-H., Goertz, C. K. and Grun,E., in Strtwtt (edited by T. Gehrri and M. S. Matthews). p.54. C:nivorsit~ of Arizona Press, Tucson. 1984.

~~ikhailovskii, A. B., 1/l.\cihiliticv of mz (~ l i r t , n i o~~ tc , r t c i i t s Pk t r tmr(7 ~~~ c~f Ph.w~o hr .stdG i i t i c , .r , Vol. 7). Consultants Bureau.New Cork, 1974.

Northrop, T. G., Dusty plasmas. P/r~x. .SC+T

high frequency drift instabilities in a dusty plasma

Documents