J. Plasma Fusion Res. SERIES, VDetermi
Nonneutral
Graduate School oflFaculty of In(200r) 550-5s4of Stream Function of 2D Vortex inand Testing Sector Probe Signals
Kiyokazu*, KIWAMOTO Yasuhitor and SANPEI Akio
and Environmental Studies, Kyoto lJniversity, Salcyo-ku, Kyoto 606-8501, Japan
Human Studies, Kyoto tJniversity Salcyo-ku, Kyoto 606-8501, Japan
ved: 5 December 2000 / Accepted: 3 September 2001)

Abstract
We present a fast
distribution from thesufficiently accurate procedure to construct the potential and the electric fieldstudies of vortex ddiasnostics withimaging is severely
dvnamics of a small

or turbulence. Using this procedure we quantitatively compare the imageng for the first time to show that core dynamics as seen clearly with theured with the probing so that its application should be limited to simple
of discrete distributions of particles.

2-D images. Such field analyses are essential for deep and extensive

Keywords:
electron plasma, sector

method, vortex dynamics, streamline

probe method. From the 2D density distribution wedetermine the potential distribution in the 2D plane. Weemploy Fourier-Bessel series expansion of the observed
density distribution for fast numerical processing. This
allows us to calculate the image charge distribution

1. Introduction
The dynamics of ais strongly magnetized andmagnetic field is equivalent(2D) Euler fluid. The corre

electron plasma thatmogeneous along thethe two-dimensional
nce between thedensity and the vorticity and at between the electric
potential and the stream fu
ease the diagnostics
x motion compared
and analvses of the electronforneutral
fluidwith those
icitv distribution are
The 2D structures of theinositv distribution ofdetermined in terms of thescreen. [1-3] Oneelectrons collected on a phos:s. however, is that it
disadvantage of this dplasma out of the trap
requires dumping of the w

induced at the surface ofthe conducting sidewall.

region. The time-varying i age charge has beenaxis. It is quite useful for in-depth analyses ofthe vortex
motions in a rotating frame.

addressed historically with atime-varying voltages induced
waves without destroying theit

extended by additional information derived from theluminosity distribution. This scheme of numerical
processing of the luminosity distribution has alsoenabled us to determine stream function as seen in acoordinate system rigidly rotating around the system

tor probe that detects
the vortex motion or

2. Experimental Configurataon

plasma. [4,5]One of the major pu

s of this paper is tocompare the probe signal
determined sienals to exami

with independently$Corresponding author's e

Our study has confirmed the reliability of thesector-probe as well as the limitation of its applicability'
and is found also that the applicability can be

The electron plasma is produced by an array ofsmall cathodes and confined in a Malmberg-trap. Il-3]

the reliability of the: n50 I I 6@ sakura.kudpc.lqoto-u.ac.ip
550@2001 by The Japan Society of Plasma
Science and Nuclear Fusion Research

Ito K. et al.' Determination of Stream Function of 2D Vortex in Nonneutral plasma

With this vacuum field configuration, the axial lengthsof the trapped electron columns are typically L = 220

ilr.e1=mm.

and Testing Sector probe Signals;aZn. n(ry' ,1. eZ{".,- e2 B;^t.(+,)

The conducting cylindrical wall in the trap region islongitudinally divided into 1l rings. The 4th, 6th and8th electrodes are azimuthally divided into 4 sectors
with a uniform separation of 90 deg. We combine thethree sectors at the same azimuthal locations and+ sinrn eL

connect them to the virtual ground of a current amplifier
through a small resistance of 300 Cl. These coupled

B;t,(?)j,(1)where J^(7^) = 0 with n = 1,2,3,...., andp";:

sectors are used as an image current detector. It is welllx ^n J ^,r(x ^)l'known that a resistive wall induces the diocorron
instability in an electron plasma. [6-8] After carefulfRtv\o, , t^1"fl' r ni'V) .
Jo)

examination and adjustment we have reached a stable
circuit configuration. [3]The transverse profile of the electron density

Q)

Here the upper limit of the azimuthal mode number m^*
and the radial mode number nMare taken as

distribution is determined from the luminosity

m*= lO0and n.-, = 100. The calculation of the potential requires

distribution on a phosphor screen to which the trappedelectrons are dumped along the magnetic field lines at
each required time of measurement. [3] The conducting

about 3 min. on a personal computer (350 MHz).

4. Testing Sector-Probe Method

surface of the screen also serves as an electron collector

Evaluating the image charge on the sector-probesurface from dQl0r at r = R, we determine the imase
current induced on the sector-probe.

to monitor the total number of electrons N. The
luminosity distribution on the screen is detected on anarray of charge-coupled device (CCD) camera. We haveFirst, we observe a sector-probe signal induced by
the single plasma string moving in vacuum. Figure l(a)confirmed a good linear relationship between N and theluminosity integrated over the screen. Using therelationship we obtain the transverse distribution of theline density N{x, y) of the plasma.

shows the time-varying distribution of a single electron

string as observed with the CCD camera at the intervalof l0 ps. The string is injected at r = 10 ps with total
electron number N = 4.5 x 107. The string rotates

3. Analysis of the lmage Data

clockwise along a circle with the radius of rs - 8.3 mm

We approximate that the chord-averaged density
n(x, y) = Nt(x, y)lL represents the actual density

at the frequency of al2tt = l0 kHz.

The solid curve in Fig. l(b) shows the current

distribution of the electrons and introduce a 2D analysis
in determining the potential distribution in the main partof the trapped region by neglecting effects from thesignal detected on the sector-probe. Here, we
approximate that the string described by the d-function.
The image current signal can be calculated easily. [3]ends. [3]The image current calculated with experimental

The potential distribution generated by the plasma
is given ina2D form @(r, 0) that satisfies the poisson,sparameters of al2n = 9.9 kH4 N = 4.5 \ 107, rslR =0.26 and sector-probe length of / = 66 mm is plotted in adotted curve. The dash-dotted line represents the output
waveform corrected for the frequency response of theequation. The contribution of the conducting wall
surrounding the plasma enters as a boundary condition
Q@, e) = 0 at the wall of a radius R.

current amplifier. We can confirm a good agreement
between the directly observed probe signal and thesignal calculated from the CCD camera image.
Figure 2 plots the amplitudes of the sector-probe
signal against N at different radii. The solid lines standfor the theoretically expected amplitude after correctionof the amplifier response. These data represent the firstexperimental confirmation that the image diagnostics
leads to results that quantitatively agree with the signals
independently detected with a sector-probe.

First we Fourier-expand Poisson equation in theazimuthal angle 0to obtain a set ofordinary differential
equations in the radial coordinate r. Q";'(r) and, n";'(r)are the mth Fourier component of the potential and thedensity, respectively. The density part nfi'(r) arecalculated from the observed CCD images. We further
expand ni;"Q) in a series of Bessel functions J.(z) todetermine the associated terms of the potential that isobtained as,551

Stream Function of 2D Vortex in Nonneutral Plasma and Testing Sector Probe Signals

ItoK. et al., Determi

Next, we compare the methods for a more
complicated distribution. Figures 3(a)-(d) show framesof the CCD images of the density distribution at---tdifferent times. Here a plasma string with Nr= 1.2 x 107
is injected at t = l0 ps into the background plasma withNt = 1.9 x 108. It is observed that the string plasma
moves toward the center of the background plasma as it
rotates while creating a spiral streak behind it. [2]/t.
la

t=20 (ps).t

I.

t=100

The solid curve in Fig. 4 represents theexperimentally observed probe signal. We try toreproduce the probe signal from the observed electron
distribution that have been recorded at a step of 2 1ts.
The calculations include the frequency response of theamplifier.
The closed circles in Fig. 4 represent the signal
calculated by differentiating the image charge
corresponding to each frame of the density distribution.
[3] Open symbols are obtained from a calculation thattakes the self-consistent drift motion of the electrons
calculated from eq. (1) and continuity ofthe density. [3]Consideration with the contribution of the vacuum fielddrive with the rigid rotation at 0 (O), 10 (A)' 15 (tr)kHz and with the differential drive (x) lead to a similar
waveforms supporting the consistency among theseanalysis. However, the image-based signals show
apparent disagreement in the phase with the probe

20't0gEo.9
a-10-20 L
0300

100TimeFig. 1 {a} Motion of the si

vacuum observed
Sector-probe signals
solid line represents thesignal. The dotted line

and the dashed-docalculated signal

string of electrons inthe CCD camera. (b)by this motion. They observed
the calculated signal

line represents thefor the frequency(b)response of the15910
oE:l.=

g50L0(ps)Fig. 2 Amplitudes of thesingle plasma string
the electron number in
R = 0.26) and closed ci
data obtained by thethe theoretical value forsignals due to then are plotted against
string. Open circles (r/lrlR = 0.121 show theSolid line represents

Fig. 3 2-D distribution of the electron density observedwith CCD camera at different time. t = 16(a), 30(b),40(c) and 100{d)ps.

case.

552

Ito K. et al', Determination of Stream Function of 2D Vortex in Nonneutral plasma andsignal.

Testing Sector probe Signals

flow pattern around a point vortex in the rotating frame.
We expect further application of this method to deepand extensive study of the vortex dynamics and statistics

The discrepancy can be reduced only by assuming
the presence of electrons extending near the wall at avery low density level unable to detect with the present
imaging sysytem. [3] Here, we assume that the invisible
distribution covers the section extending radially from r/
R = 0.8 to 0.98, azimuthally over the width of 7vl2 withthe density level of 4.8 x 1010 m-3. The electron number

rolF' /8', t.
*-AHof this part amounts to l.4%o of the total electron

Eo -18

number. Another assumption is that the electrons rotate

rigidly at the observed frequency of 15 kHz. The "probe
signal" contributing from these electrons is plotted in
the dotted line in Fig. 4. Judging from the goodproximity and total consistency, we believe that this isa-4

the unique answer to the problem posed in regard to Fig.

-84.

i/xI0I\/4080120

Time (ps)The above discussion indicates that the probe
diagnostics is strongly affected by the peripheral
distribution of particles if they exist though theircontribution in the dynamics of the main part of theFig.4 Sector-probe signal detected in the case thatplasma string is injected to the backgroundplasma as shown in Figs. 3(a)-(d). The solid line
represents the signal actually observed with theprobe. The symbols denote the probe signal
calculated from the CCD images with different
methods. The dotted line shows the signal
calculated with the model distribution extending
to the periphery. These calculated signals arecorrected for the amplifier response.

plasma is not essential.

5. Determination of Stream Function
The equipotential surfaces conform to thestreamlines. In most cases, these surfaces compose a setof almost circular lines, and fine structures as observedin the density distribution are obscured in the potentialprofile.
The streamlines to be observed in the frame corotating with the point vortex are calculated as shown inFig. 5(a)-(d) corresponding to the images in Fig. 3. [9]The density (vorticity) distribution is superposed also ineach frame. We can see a set of closed lines around thepoint vortex and the separatrix that distinguishes thebackground flow in the outer region. This structure isconsistent with the model discussed in Ref. 2 though theshape of the stream lines is strongly deviated inward.
The separatrix expands in time as the point vortex

(b)

tzv

moves inward until it reaches the center of ther/rbackground vortex to form a new concentric structure ofll,circles. It is also observed that the spiral arm extends

walong the stream lines

6. Conclusions
We have developed a fast procedure to numericallyconstruct the 2D field distribution from the image data
directly related to the electron density distribution. TheN\\\(d)fTIJAIglu

Fig- 5 Equipotential surface (streamline) and 2-D

validity and applicability of electrostaric probing

distribution of the electron density (vorticity) andat different time. f = 161a1,22(b1,40(c) and 100{d)diagnostics are critically examined by applying the fieldanalysis. It also has proved to be useful to elucidate theps.
553

Ito K. et al., Determination

Stream Function of 2D Vortex in Nonneutral Plasma and Testing Sector Probe Signals

t3l K. Ito, Y. Kiwamoto and A. Sanpei, Jpn. J. Appl.

Acknowledgments

Phys.40, 2558 (2001).

Mohri for stimulating
The authors thank Prof.
by a Grant-in-Aiddiscussion. This work was
Science, Sports andfrom the Ministry of Educat
research program ofCulture and by the collaboratince.
National Institute for Fusion

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Phys. Rev. Lett. 85, 3173 (554