J. Plasma Fusion Res. SERIES, Vol.4 (2001) I3l-137

Spiral Pattern Formation in a Rotating Cylindrical Plasma
TANAKA Masayoshi Y.* and KONO Mitsuor
National Institute for Fusion Science, Toki, Gifu 509-5292, JapanlFaculty of Policy Studies, Chuo (Jniversity, Hachioji, Tokyo 192-0393, Japan(Received: 5 December 2000 / Accepted:27 October 2001)

Abstract
Spiral pattern formation in a rotating magnetized plasma is examined. Experimental observations arepresented to show the characteristic features of spiral structure observed in an ECR plasma. To
understand the experimental results, low frequency perturbations in a rotating cylindrical plasma aredescribed using two-fluid approximation, and the eigenvalue problem is numerically solved to show theexistence of spiral solutions. It is found that dissipative drift wave instability may generate spiral patterns,
which well explain the experimental observations. The universal mechanism of spiral pattem formation is
proposed.

Keywords:
spiral pattern, rotating plasma, drift wave, vortices

1. Introduction

example spiral arms, is a function of radius r and angle
0. The radial mode is dependent on the azimuthal one,and thus the decomposition into a radial and an azimuthal modes seems to be useless. Moreover, theexistence of stationary structure means that zeroeigenfrequency solution should exist, which imposes astrong limitation on the solution of unstable modes.

Spiral patterns are commonly observed in many
rotating systems such as rotating ordinary fluids, typhoon, galaxy etc. [1]. Recently, spiral patterns havebeen observed in magnetized plasmas; Ikehata et al.
observed a persistent spiral structure in a gun produced
plasma rotating at a supersonic speed [2], while Tanaka

et al. observed a stationary spiral pattern in an ECR
plasma [3], whose rotational velocity is in subsonic

Then, the problems of interest in this study are
whether or not such a particular solution does exist, andhow it makes spiral pattems.
In this paper, we show that dissipative drift wave
instability in a rotating magnetized plasma can produce
a stationary spiral structure in density and potential. In
the first part, the experimental results on spiral structure

region. These experimental results suggest that spiralstructure is a robust entity, and the underlying
mechanism on spiral formation is common among manydifferent systems.

It is easily understood that self-generation ofcoherent structures originates from a certain kind of
instability in the system under consideration. Then, we
expect that the observed spiral is a developed state of
instability grown in a rotating magnetized plasma, andthat eigen-mode analysis under the given boundary
condition should be done as usual. However, describing

observed in an ECR plasma are presented, and it isshown that the observed spiral is a kind of density
modulation pattern categorized into an Archimedeanspiral. Then, to understand the experimental results, lowfrequency instabilities in a rotating magnetized plasma
are described using two-fluid approximation. The linear
eigenvalue problem is numerically solved to show the

spiral pattem seems to be difficult, because the locus of
constant phase points in the polar coordinate system, for*@2001 by The Japan Society of Plasma
Science and Nuclear Fusion Research

C o rr e sp ondin g autho r' s e - mail : my t anaka @ nifs. a c.j p131Tanaka M.Y. et al.. Soiral Pattern Formation in a Rotatins Cvlindrical Plasma

existence of spiral solutions, which well reproduce thecharacteristic feature of experimental observations. The
universal mechanism of spiral pattern formation is
proposed with a simple example, and it is concluded thatspiral structure is a kind of interference pattern between

the experiment, and the electron temperature and density

azimuthal mode function and radial mode function, thelatter of which becomes complex in an unstable system.

Iength.

were 5 eV and I x 1012 cm-3 for the input power < I

kW, respectively. Stationary spiral patterns wereobserved in the pressure range |-2x 10-2 torr. The sizeof plasma was 20 cm in diameter and 50cm in axial
Figure I shows an end view image of a spiralpattern taken by a CCD camera located at the other open

2. Experimental Results

end of the cylindrical chamber. This pattern doesn't
move with time, when the experimental conditions are
kept constant. This means that the eigenfrequency ofthis pattern is equal to zero. As seen in this figure, there
are two winding arms, the radius of which in the polar
coordinate system is proportional to the angle of rotation
as shown in Fig.2. This is the characteristic feature of

Plasmas were produced in a cylindrical chamberwith external magnetic fields, by a microwave with afrequency 2.45 G}iz launched along the field line froman open end of the chamber. The magnetic field
configuration was so-called a beach structure for thismicrowave, and the electron cyclotron resonance point
was located in the middle of the chamber. An electron
cyclotron wave was excited in the plasma [4], and fully
absorbed before reaching the ECR point because of thelarge absorption coefficient. An argon gas was used inArchimedean spiral.
To identify what makes spiral structure, we carriedout visible line image measurements (see Fig. 3). Figure

-t80

-120

-60

0

60

120

180

0 (degtee)

Fig. 1 End view image of spiral structure in an ECR
plasma.

Fig. 2 Arm radius as a function of angle in polar co-ordinates.

Fig. 3 Line emission image. Left: Arl l42O nm), right: Arll (488 nm).

r32

Tanaka M.Y. et aI-, spirar pattern Formation in a Rotating cylindrical plasma3(a) is the gray scale image for neutral argon line ArI(420 nm), and Fig.3(b) for argon ion line ArII (4gg nm).

the outer region, indicating existence of shear (thedirection of rotation is clockwise for Fig. 3).
When the magnetic field is inverted, the polarity ofrotation in spiraling into its center (refer to as polarity ofSince the electron temperature is almost constant over
the whole cross section, the gray scale in Figs.3(a) and(b) are roughly proportional to nsn. and nin",respectively. The same patterns are seen in the both
figures. Considering that the pattern of ArII image

spiral in the followings) changes the sign as shown in
Fig. 5. It should be noted that the same patterns are seen
in both figures except the polarity of spiral. Since theplasma rotates due to E x B drift, and this velocitychanges the sign with the inversion of magnetic field B,
the result suggests that azimuthal rotation of plasma
plays an important role in exciting the spiral density

represents density perturbation, we can conclude that theobserved spiral is a density modulation pattern.

As mentioned above, the spiral pattern doesn't
move with time (stationary structure), however, thebackground plasma rotates around the center axis due toperturbation.

E x B drift. The plasma flow velocity was measured
with a directional Langmuir probe [5], and theHere we summarize the experimental results;(l) Spiral pattern is observed in an azimuthally

normalized azimuthal velocity as a function of radius isshown in Fig.4. As seen in this figure, the plasma rigidly

rotating plasma.

rotates in the core region (r/a < 0.5, a: plasma radius),
and then the azimuthal velocity decreases with radius in(2) It is identified as a density modulation patternand categorized into Archimedean spiral.

(3) The spiral pattern does not move with time,
which means that the eigenfrequency of thisstructure is equal to zero in the laboratory
frame.

(4) The background plasma exhibits an E x B
rotation, and there is a shear in azimuthalo.3

velocity profile.

c>"

S o.z

(5) The polarity of spiral winding changes its signwhen the direction of magnetic field iso.1

inverted.

It is well known that rotating fluids excite various
types of instability, and generate spatio-temporal
structures such as Taylor vortices. According to the

o.o

o,0

0.2

0.4

0,6

ila

experimental observations, it is natural for us to consider

Fig. 4 Azimuthal flow velocity profile.

electrostatic instabilities in a rotating plasma withBO

B@

Fig. 5 Gray scale images of spiral structure. The direction of magnetic field
133

is inverted in the right figure.

Tanaka M.Y. et aI., Spiral Pattern Formation in a Rotating cylindrical Plasma

thermal velocity. The quantities C)i and Q" are the ion
cyclotron frequency and electron cyclotron frequency,
respectively. The second term in ion velocity equation isthe correction due to centrifugal force. Other velocity

external magnetic fields.

3. Low Frequency Perturbations in a
Rotating Magnetized Plasma

We consider low frequency perturbations in arotating plasma with external magnetic fields. Special
attention is paid to stationary structure and its spatial
pattern, to explain the experimental observations
Low frequency perturbations in a rotating plasma is
formulated using two-fluid approximation [6]. The

continuity equation and the equation of motion forcomponents for both ions and electrons are also
obtained from Eq.(2).
Substituting these velocities for both ions andelectrons into the continuity equation, linearizing andinvoking the charge neutrality, we have the following
equation for the perturbed potential Qa with a mode
number I :rY.v(n@
E''o'

dt

(l)

a@'))=o ,* t)@t .v t)td)- =:n(-ro*c] ,'"'*tr lI
m"\
Yp"
u@) ,
- 6""
"'' =lffoffio " -ro

t

1

Q)

where d stands for ion or electron (i or e), n(d) thedensity, and u(o) the velocity. We take into accountdissipation due to ion-neutral and electron-neutral
collisions, for which vd means the collision frequency.

d ln nodEd€ a d4'

a)in Eq.(7); f1(ar) = vi-i(a- (' a[\1. and l.(ar) = v.-i(a- t ,3\. This equation describes low frequency

be determined in a self-consistent manner' we here
assume them, for simplicity. AU the physical quantity

instabilities such as dissipative drift wave instability'
centrifugal instability and Kelvin-Helmholz instability[7]. The contributions from flute modes (centrifugal
instability and Kelvin-Helmholz instability) are muchsmaller than that of dissipative drift wave instability

AQ, z, A is decomposed into the unperturbed part' andperturbed one, which can be expressed by the radial,azimuthal. and axial modes;
te - ot1* dq8' * t d'u';bwhere ars and 0)* are the angular frequency of E x B
drift and diamagnetic drift respectively, and { is thenormalized radius rla. The following notations are used

The quantity { is the electrostatic potential. The ions are
assumed to be cold, and the pressure gradient force isconsidered only for the electrons. Although the profile
of unperturbed quantities, ns(r), as(r), and @6(r)' are to

A (r, z, o) = Ane) + A1 (r)' si(kz+

I

^
"..|
o'1,'od'9*l+*=0, (6)lY*lBte>{lo,d€' '11' d4 I d6 l', '"', ?',|"-k2o,f)" l. - (ot. \P \5 ) =F-,tal r1 rarl \' a -V r,electrons and ions are written bY ,(3)electrons to E x B drift and diamagnetic drift. The
stationary azimuthal velocities for ions and electrons

under the present experimental conditions, and thus weconcentrate our attention to the dissipative drift wave
instability (the first term of Eq.(7)).
To obtain the stationary solutions, we first fix thereal part of eigenfrequency ar to zero, and numericallysolve Eqs.(6) and (?) with free parameters of imaginary
part of ar and wavenumber ft. For this purpose' Eq'(6) isdue to the unperturbed potential and density aretransformed into a set of ordinary differential equations,where k is the wavenumber (complex) in axial direction. L the azimuthal mode number, and ar the eigenfrequency.

The ions are subjected to E x B drift and the,s,=:ql,-r*+l'+'g"l-ou ol+vf
r dr[' \o]+vl/,[?=ffiS(o.t l-tnno(r))

and then numerically integrated by the Runge-Kutta
method, in which the boundary conditions(4)

Qt=O at €=O (center),
Qt=O at €=l (walt)

I

(5)

(8)

are imposed.

Figure 6 shows the radial eigenfunction @a for A =2 mode, in which the solid line indicates the real part'

where C5 is the ion sound velocity, utr, the electron
134

Tanaka M.Y. et al., Spiral Pattem Formation in a Rotating Cylindrical plasma

and the broken line the imaginary part. It is noted thatthe radial eigenfunction becomes a complex function,and there is a phase difference between the real andimaginary part. The phase difference plays a key role inproducing spiral patterns, which will be discussed in

0.10

later section.
Two-dimensional pattern is visualized by making a

0.05

product ot Qp(r) expli(.01 and its contour plot for rhe
real part, which is shown in Fig.7. A spiral pattern with
two arms is clearly seen in the figure. Since the density
perturbation is related to the perturbed potential, thespatial pattern of density perturbation contour is thesame as Fig.7. The relation between arm radius and its
angle of rotation in the polar coordinate system indicates

l*"

:

o.oo

o

rlro

that this spiral is an Archimedean spiral.

Fig. 6 Radial eigenfunction (Ref.6).

The vector field plot of flow velocity associatedwith this spiral structure is depicted in Fig.8 In thisfigure, each arrow is indicated by dark gray for positive
radial velocity and by clear gray for negative radial
velocity. It is interesting to note that the flow field
associated with this spiral structure is not a sink nor asource; the plasma goes in along the clear gray arrows,
reaches the center, and then goes out along the dark grayarrows, in other words, the flow vector field associatedwith this spiral is a circulating structure between thecenter and peripheral regions. This flow vector field iswell explained by the E x B drift of the perturbed
potential presented in Fig.7.

4. Universal Mechanism of Spiral Formation
We consider here the mechanism of spiral patternformation. As already mentioned above, the existence ofFig. 7 Contour plot of potential (Ref.6).

complex radial eigenfunction is essential in producing
spiral pattern. Let us take a radial function Q(pr), andmake a product:

Q(Fr).exp[il0l(e)where B is a parameter. When Q$r) is a real function,
the radial and azimuthal structures are independently

determined by the radial and azimuthal functions
respectively, never generating a spiral pattern. However,when @(Br) becomes complex, this product is rewritten
as
| 0(B

r)l' exp [iArg[ / (F r)] + i I el

(10)

The argument of radial function ArgtQ(Br)1, which is
generally a function of r, is renormalized into the phaseof exponential function, and produces an r-dependent
phase. The condition of constant phase:
Fig. 8 Vector field plot of flow velocity associated withAreIQ(Fil + [ 0= s6ns7 .

spiral structure (Ref.6).
135

(1 1)

Tanaka M.Y. et aI., Spftal Pattern Formation in a Rotating Cylindrical Plasma

5

0

-5

-ilt

-o-50510
x

Fig.9 Contourplotof aproductof Bessel functionandexponential function.(left:p=1.0;right F=1+O.25ildriven by both the E x B drift and the diamagnetic drift.
The difference in azimuthal drifts between the ions andelectrons induces charge separation, which cannot be
fully neutralizedby the electrons whose axial motions
are dragged by the collisions with neutral particles. Thus

gives an r - 0 relation for the locus of constant phase
points, producing a spiral nature in its spatial structure.
Archimedean spiral corresponds to the simplest case, inwhich the argument ArSIQ$r)l is proportional to r.
We give here a simple example. Let us take aproduct of Bessel function and exponential function
J(Br).eito ( (, = 2). When the parameter B is real, thedrift wave instability takes place and the density
modulation associated with this instability is organizedin such a way that the central part rotates almost rigidly,
while the outer part lags behind the central part becauseof the shear in azimuthal velocity profile, consequently

azimuthal pattern produced by this product iscompletely determined by the exponential function, as
seen in Fig.9(a). However, when B becomes complex,
an "interference" between radial and azimuthal mode
functions takes place, and the contour plot of thisproducing a spiral structure.

Although we have reported the occurrence ofstationary spiral, which is easy to find out in the

product generates a spiral pattern as shown in Fig.9(b).
It is emphasized here that we don't assume any specificphysical situations so far. Therefore, this scenario on
spiral pattem formation may be a common mechanism

experiment, the occurrence of non-stationary spiral will

be more general and appears in a wide range ofexperimental conditions.

underlying in many other systems [8,9].

5. Summary

To give the physical meaning of complex radial

The experimental results and theoretical analysis on

eigenfunction, we return to Eq.(6). When B in Eq.(6) isreal, the radial eigenfunction becomes a real function. A
complex radial function occurs only when B is complex.
Noting that the complex B is due to the system being

formation of spiral structure are presented. The spiralpattern in an ECR plasma is identified as a density
modulation excited in a rotating magnetized plasma.
Linear mode analysis on the low frequency instability
shows the existence of stationary spiral solutions, which

unstable (dissipative drift wave instability), we canconclude that the occurrence of spiral nature is
attributable to the instability. It should be pointed outthat instability plays an essential role in two aspects: (i)well reproduces the experimental observations.
Since the present analysis is a linear analysis, wecannot determine the absolute value of the amplitude of
perturbed density, which should be balanced, in thecourse of instability development, with the higher ordereffects and the nonlinear effect. It is very difficult todeal with this problem for general cases. However,when the characteristic scale length in radial direction ismuch smaller than the axial one, these scales are
separable by using the reductive perturbation method

self-generation and growth of structure, and (ii)generation of spiral nature.

The physical picture of spiral formation in thepresent experiment is the followings; the ions rotate
azimuthally by E x B drift. Since the ions are muchheavier than the electrons, they are subjected tocentrifugal force, the rotation frequency is affected bythis effective sravitational force, while the electrons are136

Tanaka M.Y. et al., Spiral Pattem Formation in a Rotating Cylindrical plasma[10]. In that case, we obtain Eq.(6) for the radial
eigenfunction, and the one-dimensional GintzburgLandau equation for the axial eigenfunction, which
permits localized solutions made from competitiveeffect between the dispersion, diffusion, and thenonlinearity [1]. The existence of spiral structure,
which is axially localized with finite amplitude, is

t5l K. Nagaoka et al., J. Phys. Soc. Jpn., 70 131

possible.

[7] A.B. Mikhailovskii, Theory of Plasma Instabilities(Nagoya, Japan) Vol.2, 1650.

t4l M. Tanaka et al., J. Phys. Soc. Jpn., 60 1600
(1991).

(2001).

t6l M. Kono and M.Y. Tanaka, Phys. Rev. Lett., 844369 (2000).

Spiral structure is a kind of "interference " pattem
between azimuthal and radial functions of unstablesystems. This simple mechanism may universally occur
in other unstable systems.

(Consultants Bureau, New York, 1974), Yol.2,
Chap.1.

t8l J.R. Penano, G. Ganguli, W.E. Amatucci, D.N.
Waliker and V. Gavrishchaka, Phys. Plasmas, 54371 /1998]..

t9l A.T. Burke, J.E. Maggs and G.J. Morales, Phys.

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t37