J. Plasma Fusion Res. SERIES, Vol.2 (1999) 331-334

Spiral Structure Formation in Rotating Plasmas
KONO Mitsuo and TANAKA Masayoshil
Faculty of Policy Studies, Chuo University, Hachioji, 192-0j93, JapantNational InstituteFusion Sciences, Toki, 509-5292, Japanfor(Received: 9 December 1998 / Accepted: 13 May 1999)

Abstract
A theory on spiral structure formation in rotating plasmas has been formulated and shown toreproduce the characteristic features of spiral structures observed in electron cyclotron resonanceplasmas.

Keywords:
spiral structure, rotating plasma, centrifugal force instability, collisional drift wave instability

1. lntroductionu-u"Organized structures in plasmas have been a topic
since they may not only give a deep insight into general
aspects of self-organization in complex systems but alsoot

* ro' Vuo =fiero*Lr'xB)-Ior"-vouo.

Q)

contribute to transport phenomena. In fact recently
coherent structures have been observed in laboratories

where no and uo are the density and velocity of particleof species a (electron/ion), and p and S are the potential[1-4] and are subject to theoretical analysis forand the source due to ionization, respectively. In thefollowing ion temperature p1 is assumed zero.
A potential in magnetized plasmas in a cylindrical
vessel is determined by the balance between ion crossfield diffusion and electron axial transport sinceelectrons are magnetized almost completely while ionscan diffuse in radial direction. The space potential
produced by the ion radial transport is short-circuited by
the electron axial transport so that we have V(n[u[) = V(n6ud) = Sn6 with the background density nff andunderstanding underlying physics. Both stationary [3]and rotating [4] spiral structures are observed.
In this article we have developed a theory toexplain the formation of spiral structures in magnetizedplasmas and showed a good agreement of the resultswith observations in ECR plasmas whose characteristicfeatures are (l) the direction of the arm stretching isreversed when the magnetic field is reversed, (2) thestationary'structure is observed only in a narrow rangeof the background pressure, and (3) the spatial extent of
the structure in the axial direction is of the order of theplasma radius.

velocity uff, which gives the relation between the densityand potential. In the following we look for a potential

2. Basic Equation

profile for a given background density since theambipolar equation is not easy to solve except for(1)

special cases. Invoking charge neutrality n6Q, z) = n6?,z) = no(r, a) the ambipolar equation gives the potential
QoQ, z) which can be expressed in terms of theI author' s -mail : kono @fu s. c huo u. ac.j p@1999 by The Japan Society of PlasrnaScience and Nuclear Fusion Research

Equations for magnetized plasmas readff +Vfn"uo)=Sn",C orre spondine-331

Kono M. et al., Spiral Structure Formation in Rotating Plasmas

background density no?, z) with normalizations ( = rlyo
and 4 = zlrawhere 16 is the plasma radius, as

QoG. ry=_e.v y2('iiwheref "1ot1=v"-i1a- {a"o;,tn no{€.4)Z

alsF, |
[e'- 4 J,f "'"y,pot€'nJ(. DJo "td2no(€"' n)3,,.ry\+tl.

l€no('e" -'where t = ,i(#-"1ta]= ()"Y I + 1"1a42.
- 2a)E) 6) " - 2o16 - E'de

Substituting these into the equation of electron
continuity we have'(3)ltp-frn*sl4'noj ,t. and we have assumed that (y"/t=u11,@)2,1oty'r2oQ^t2 << I .

The rotation frequencies of ion and electron+t1+14tn
L,a,-!1,'q' ' no '-&- lr dE

dE E"(a)'',(' ,i( z,(a\ , 6 nn(Q"-2a6\.,-'r-E

azimuthal drift are given by.aE@[= ,d!(J -Ar),

aeo= as- (D.,d'- t6, - 41' no'A€, "(4)"+^t(0,-!))++

where oJs and al- are the E x B rotation frequency
and diamagnetic drift frequency, respectively defined byn0 "d6- tJa)

3- =+-'l

1,0,-5.
'rf,nn d4 f"1ay d4

noOn the other hand the fluctuating parts of ion velocity
are given in a similar way as before by

C: +-+.
taO A & +---"
ldlnr^
Ac=t' ,1Q,8
dq- ' =-----LriQ"€ d€| 6)t+2a'ilro, + r,rcndUl\r'rr\w' a€ t'€
c:aLt,d@b-U Ii{, =i;@)L(Qi+2a'o+€,i"t =-The difference of the ion and electron rotation
frequencies is an origin of instabilities, a centrifugal
instability and a dissipative drift wave instability which'! -iroz,qrol',-support spiral structures.

0,,.
-!'L'''q

3. Spiral Structure of Fluctuations

C:,W)=---' ,uf,1.lo1 d4

The fluctuating parts of the electron velocity 0(r, e)

in a reference frame of normalized cylindricaldQt

coordinate (8,0, nl which are expressed asr!'nt\/ ':u.(r. z)= =. , (8.,4)+ | (€,'D le',"whererj(o)=v,-i(a- loti),:rA\*v.

2,(a)- (9, + 2r;;i) (9, +2ai.,. 6#1+ r,1a42.

l

are given from eq.(2) byui =Substituting these into the ion continuity equation wehavei(tA"_2ai.l (Q' niuI- ;)ror"-.,1- --*ftt2, =-#[(r2"'ro2"114)1''ni[i(a- Zati')+S],n+r-@t
l,a, -!fno-t.
- dE
A (0, n,J
2.:o3- "d0"ot A n^_c'z,f Jal I d, 6,.-tE*('rlz,1,,1i€tn2i@)no(Q;-2a6\.,,(' il,drS - no )- d€'d6"'" C-:*i5!1,3 ta,-a6,t6ttla4

f -E ,rrJ.,ltTr- ,^^td,no3o,,.
_g:-,ino6i'f,r,,dq''''

^ ,I
ttvr

-L

(T"@t,ta," -[tl.
no-,i='1"'- rnf,1ro1d4\YIno)'oQ,

Invoking charge neutrality ni - ni = n,. we have332

Kono M. et aI., Spitral Structure Formation in Rotating Plasmas

A2&,| dlnr?^ aThe main contribution comes from the collisional drift

12ro*ri* it66Qt -wave instability for r - I and from the gravitational
instability for r - 0. For r - I the axial extent of theplasma is of the same order of the plasma radius, whichis the situation of the ECR plasma in which spiralea,(a"| trzr^*:d'b-i!i r,lrtty'- ;E * ,- lt.'

19*tY*64t0,a€ -dE'd€'"'structures are observed. Then the solution isapproximated in the case of weak {-dependence of thezero-th order quantities byA? Dt v'* - Q"Q,
* rr.JqrJro) rla) an,drt
a2rrvr (€)= Ip(F0,*:]*tnt =0. (s)where Ja is the Bessel function of the first kind. The
real part of the argument of the Bessel function should

be positive to give a convergent behavior while theimaginary part is responsible for a spiral structure whichcomes from ll(ar) and |(al) with co = co, + iy .

We have neglected terms of the order of or less than O(ft1 anO used the following approximation which isvalid when the scale length of the fluctuations is smallerthan that of the background density,

ntlQ,u,,^=-(e)

Multiplying tyj to eq.(1) and integrating theresultant equation from the center to the edge of thel dlnno ^plasma under the boundary conditionrr^ar,r-(@6*r)E a4 ''=
lax
" Qr.
o- (aEVt (0)= Vt G)=0,(6)we have

Equation (5) describes low frequency fluctuations
subjected to both a flute mode instability due tocentrifugal force acting on ions and a collisional drift

ft -,.dv{.,+:lvil'12 , ,,7'<€)lvrl' j d6=0. (10)€l+f|Jodg E"

wave instability, though modes related to spiral

From the imaginary part of this equation we have for r

structures are decaying instead of propagating along the-lfa,n ( |,A?lvtt$l' d?= ( <al'o>.

axial direction.
Assuming the axial dependence of the potential as

Qt(€,D=Qr(€)ewhich guarantees that the real part of F'(0 i" positiveand the imaginary part is proportional to O;, indicating
the winding direction of spiral arms is reversed when*',and putting

Qt (5)=

the magnetic field is reversed. Furthermore a stationaryspiral structure is formed when a so called Rayleighvr (€)
I noG)

condition

we can transform eq.(5) tod'v,.ldv'-#a,=(106>=0,02

iE.t'F r€t-itv' =o'can be satisfied, implying there is a radial point where1116 = 0 and electrons cannot locally cancel the charge
separation produced by ions which rotate with ari - ars.

a)

where

For a wide range of parameters, however, spiralstructures are rotating.

For r - 0 we cannot have a stationary spiralfo,structure since when ( ,oB) = 0, the real part of B2G)
also becomes zero and y,2 is expressed by a linear

K-1,combination of (-( and (t,that is, y4 is divergent.
In the following we compare our results with theexperimental data in the ECR plasmas [3]. A stationary
spiral structure is shown in Fig.l for which thebackground density is chosen as no(6) = exp (-aE2 blo), (a = 312 and b = l) and is compared with the(8).fo,K- 0,333

Kono M. et al., Spiral Structure Formation in Rotating Plasmas

0-Bv

o.7
0.

0.2

-0.25

-0.75
-0.8
-0.8;<0.8

Fig. 1 A spiral structure: the radial profile of the real and imaginary parts of eigenfunction Vr (El leftl and the contour ofan eigenfunction (right).

experimental data in Fig.2. The corresponding ion
azimuthal drift velocity is compared with the observednO(E) rone in Fig.3.

n4. Discussion
In this paper we have developed a theory on spiralstructure formation in rotating plasmas and reproduced
the characteristic features of spiral structures observedin electron cyclotron resonance plasmas[3]. The energy

0.

0.2

0.4

0. 60.8E

1stored in a velocity shear is released to give aninstability which drives a spiral structure. Under theFig.2 A density profile given by no(6) = exp (-aE2 - bCal
with a = 312 and b =1 (a solid line) and thespecial condition that there is a radial point where arf =0, a spiral structure becomes stationary which has been
observed in the experiment [3]. Beside stationary spiralobserved data (a dotted line).

structures there can also be excited rotating spiralstructures for monotonically decreasing potentialvoi(E)suuctures.

0.75
0.5
0.25

References

tll M.V. Nezlin and E.N. Snezhkin, Rossby Vortices,
Spiral Structures, Solitons, (Springer-Verlag,Berlin, 1993).

t2l K.S. Fine, A.C. Case, W.G. Flynn and C.F.
Driscoll, Phys. Rev. Lett.75,3277 (1995).
t3l M.Y. Tanaka, T. Sakamoto, H. Imaizumi, K.
Taniguchi and Y. Kawai, Proc. 1996 Int. Conf. on
Plasma Physics (Nagoya, Japan) Vol.2, 1650

-0.
-o.75
0.5

(1997\.

t4l T. Ikehata, H. Tanaka, N.Y. Sato and H. Mase,Fig. 3 An ion drift velocity: theory (a solid line) andobservation (a dotted line).

Phys. Rev. Lett.81, 1853 (1998).
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