J. Plasma Fusion Res. SERIES, Vol.4 (2001) 124-l3O

Controlled Experiment on Two-Dimensional
Vortex Dynamics in Electron Plasma
KIWAMOTO Yasuhito*, ITO Kiyokazur and SANPEI Akior
Faculty of Integrated Human Studies, Kyoto (JniversitySalqo-ku, Kyoto 606-8501, Japan
IGraduate School of Human and Environmental Studies, Kyoto University, Sakyo-ku 606-8501, Japan(Received: 5 December 2000 / Accepted:27 August 2001)

Abstract
Fundamental processes in 2-dimensional turbulence are studied experimentally in terms of vortex
dynamics in an electron plasma on the theoretical basis of the isomorphism between the Euler fluid andthe guiding-center electron fluid. The initial distribution of the vorticity is controlled precisely in a wide
spectrum of shape and strength by using an array of small electron emitters. By employing imaging
diagnostics we examine key elementary processes at work behind the successive appearance of quasiequilibrium "crystallized" states of decreasing number of discrete vortices. The observation includes
dynamics of a single or two discrete vortices interacting each other or with an extended backgroundvortex, and structural deformations of the background vorticity distribution.

Keywords:
electron plasma, vortex, 2-dimensional turbulence

diagnostics includes dumping of all electrons along themagnetic field lines to an Aluminum-coated phosphor
screen that also serves to electronically determine thetotal charge of electrons -eN. The linear relationship has

1. lntroduction
A set of equations that describe 2-dimensional (2D)
motions of guiding centers of electrons transverse to theexternally applied homogeneous magnetic filed B6i are
isomorphic to those of the 2D Euler fluid of noncharged particles t 11. The relation 6 = en"ltnsBo = oirl a,between the electron density n.(x,y) and the vorticity((x,y) = (2 = Y x y makes the diagnostics of the flow
distribution with y = E x B 6l fi = Q x Y Ql 86 much easier

been demonstrated between /(-t,y) and the lineintegrated electron density distribution Nr(r,y) =tdzn.(x,y,z) in a wide dynamic range of 1 : 3000 with ahigh sensitivity of I count per 10 electrons [2]. In theMalmberg-trap configuration employed for thisin the electron plasma. Here rr4 and ro" are the local

experimental study, the typical axial length of theplasma and cyclotron frequency, respectively. Since theelectron plasma is L = 235 mm over the cross section so

stream function r4, defined by v = 2 x Vt4, is related tothat n"(x,Y) = Nlx,Y)lL [3-51.

A new feature we have introduced to the arena of

potential distribution simply by @(-r,y) = BoV@,y), theelectrostatic analyses are directly applicable to the flow

vortex dynamics researches is an array of small electronemitters each of which can be operated independently.
Various configurations of electron strings (point vortices
in 2D space) can be generated as the initial distribution
or added sequentially [3-5]. Point vortices have playeddynamics.

The distribution @(x,y) is obtained numericallyfrom n"(x,y) that is determined experimentally in termsof the 2D luminosity distribution /(x,y) on chargecoupled-device (CCD) camera images [2]. This*

Corre sponding autho r' s e-mail : m52279 @ sakura.kudpc.lcyoto-u.ac.jp

124@2001 by The Japan Society of Plasma
Science and Nuclear Fusion ResearchKiwamoto Y. et al., Controlled Experiment on Two-Dimensional Vortex Dynamics in Electron plasma

significant roles in researches of 2D turbulence both

Here the first term stands for direct interactions withother vortices, and the second term comes from image

with dynamical approaches and with statistical
approaches [6]. Different shapes and levels ofcharges induced on the conducting cylindrical wall at l7l

continuous vorticity distribution can also be generated= -R which conforms to the outermost stream line r4(R).
The circulation fp of vortex B is related to the electron
density as follows:by combining repeated injection, relaxation and partial
dumping of trapped electrons. [5].
In this paper we report some fundamental processes
that is involved in the evolution of 2D turbulence bystarting with controlled initial conditions of vorticity
distribution. Some other topics are presented in separate
papers in this conference [7,8]fflB= f f dx,dy,((x,,y,)
rIt^
J JJA
P

ff

=on"(t',y' \eN B

JJ'u*'o''-zet-ffi" (3)Here Sp is the cross-section of the vortex patch B
consisting of Np electrons. The frequency e(zl)2, Discrete Vortices in Vacuum
The 2D location (x,y) of a point-like discrete vortex
a interacting with other point-like vortices in the trap ofradius R is described in the complex plane with z = x +iv as.

represents the external drive that is generated from theradial electric field associated with the end-plugging
potential;

Experimental confirmation of the Hamiltonian
dynamics as represented by Eq. (l) has been carried out#=W,""-,n(l ,"1)rL,

for small number of point vortices [3,4]. Increasing theinitial number of vortices introduces stochasticity in theevolution of the point vortex dynamics, and leads to(1)wherew(z)=punctuated merging among discrete vortices [9].

E"*^(*-t)_F- f"/-*'"(,n_*)One example is shown in Fig. 1. Here the initial
distribution of 19 vortices goes into a random state ofmotion until it reaches a meta-equilibrium state of 8vortices. Before all the vortices merge into the single-(2)0 FSGG

3m0LolFig, 1 Evolution of interacting discrete-vortex system to ordered states.

r25

Kiwamoto Y. et aI., Controlled Experiment on Two-Dimensional Vortex Dynamics in Electron Plasma

185

Smsec

{-€

30.6mm20m

LolHiFig. 2 Evolution of interacting discrete-vortex system to ordered states. The radial potential profile at the ends is slightlyflattened.

vortex state, there appear several stages of meta-appear significantly to affect the process of interactions

equilibrium configurations. Such crystal-like states havealready been observed to evolve from random
distributions of vortices created by Kelvin-Helmhortz
instability under the magnetic field stronger than ourcase by a factor more than l0 t9l. Mildness in theprofile of each vortex is most probably attributed to theweakness of the present magnetic field Bs < 0.048 T.
Figure 2 shows another observation that slightflattening of the radial profile of the vacuum potential at
the plugging zone leads to different configurations of
the vortex crystal that has inner ring consisting of 4 or 3

among remaining vortex Patches.

(later) vortices. The difference in Fig. 2 may beprocesses in planetary atmosphere, and qualitative
behavior has been examined experimentally with normal
fluids. Quantitative examination became available only
after Schecter and Dubin proposed a theoretical model

3. Single Vortex Dynamics in Background
VortexAs an important elementary process we examine
the dynamics of a single vortex in interaction with abackground vorticity distribution [3-5]. In the shorthistory of our experimental studies this type ofinteraction came out as a basic problem that is
simplified from the two-vortex interaction as discussed

later. This configuration is among well-discussed

attributed to a reduction of the radially-increasing
azimuthal-drive due to the radial electric field of theconfining potential. This vacuum-field drive tends tobased on linearized Euler fluid equation of motion [10].
The radial velocity of a point vortex projected to thecompensate the rotational shear driven by the self-fieldof the electron vortices.
In the process of the relaxation relatively weak
strings are broken into thin sheets, and get folded into astronger vortex coming nearby. A noticeable process
involved here is that some fraction of the sheet remains
unabsorbed and forms a rugged background vorticitydistribution. The self-generated background vortices

gradient of the background distribution is given in termsof differential equation,#=*tr"(r)-"-(+)'
t26

(4)

Kiwamoto Y. et al., Conlrrolled Experiment on Two-Dimensional Vortex Dynamics in Electron plasma

Fig. 3 Motion of a single point vortex in the background vorticity distribution. From Ref. 5.

where A = -rd{2*r)/dr is the shear relative to the local

background. The level of the perturbation increases withthe strength (* N,) of the point vortex. Observations
indicate that on the arrival of the point vortex the local
perturbation level goes up to several tens ofpercent andrigid rotation

Clr{r)="--[€s6or' Jo

drrn,(r)stay at this level even after the vortex moves away.

of the background vortex, t = (f ,l2nlAl) r/2 is the radiusof the stagnation zone around the point vortex with acirculation of f,, and c is a number of order l
In short this model predicts that a point-like vortex
moves up along the gradient of the background vorticity
with an extended distribution. The direction of the
motion of the point vortex has been confirmed for
different shapes of the background vortex including
convex and concave (unstable) radial distributions [5].
Though the fluid dynamical expression appears highly
mathematical, it turned out much easier to understand if
we interpret this process in terms of plasma physics [5].
Figure 3 displays snapshots of the vorticity
distribution after a single vortex string is placed at the

Figure 4 shows the time evolution of the radial
distance of apoint vortex from the center of the
background vortex for different strengths of fu * fy',
where N"/106 = 1.6 (A), 4.4 (!), 8.7 (O), 17 (O)[5]. The electron string is electrically disconnected fromthe cathode at the time t = 15 ps as indicated by thearrow. The inset shows the common profile of the initialbackground in unit of 1013 m-3 with N6 = 7.8 x 107. The
arrow here indicates the initial position of the point
vortices. The curves attached to the dots denote theorbits as calculated from eq. (4) by using experimentally
determined parameters. Well before the point vortexreaches the background center, the observed radial orbits

agree with the theoretical expectation as given by

periphery of a background vortex with a radialsolidcurves.

distribution that decreases monotonically. The upper

The characteristic time, required for the point

panel shows the total distribution, and the lower panel

vortex to travel between l0 Vo and90 Vo positions alongshows the perturbation in the background distribution[5]. The point vortex moves radially up the hill of thebackground while orbiting azimuthally around it under
the drive of the .B x B drifr. A spiral streak of thevorticity perturbation is generated behind the point
vortex and remains long after it reaches the center of thethe initial path of the radial distance from thebackground center, is evaluated from the experimental
data as shown in Fig. 4, and plotted in Fig. 5 as afunction of ?yo6"1. The the oretically expected time 4n*.,

is evaluated from eq. (4) by introducing theexperimental parameters including the initial profile of127

Kiwamoto Y. et al., Controlled Experiment on Two-Dimensional Vortex Dynamics in Electron Plasma

imaging diagnostics, we have constructed 2D
distributions of the potential and electric field [2]. It isshown that the spiral structures in the vorticity
distribution actually generate azimuthal electric field at
the position of the point vortex and can drive it radiallyin and out at the velocity consistent with that evaluatedfrom Fig. 4 [7].
The discussion in this section indicates that thedistribution of background vortex is modified

1030

substantially by a discrete vortex with high local density
and the modified distribution exerts influences back tothe latter.

50

Time (ps)Fig. 4 Radial distance of a single point vortex from thebackground vortex center.

4. Two Vortex Interaction in Background
VortexAs an elementary process in the interaction among
discrete vortices, we examine trajectories of two discrete
vortices immersed in different levels of background
vorticity. Example frames of the vorticity distribution
are shown in Fig. 6. In contrast to the dynamics invacuum, the two vortices approach in a time shorter byorders of magnitude either to merge into one or to forma binary vortices. The binaries revolve stably at adistance of less than two vortex diameters at which theyshould get merged in vacuum [1].

110.

Tilod.r (psec,As depicted in Fig. 7 that was obtained in thepreliminary stage of our study [4], the time t required
for the two vortices approach decreases almost
proportionally to the inverse of the total circulation f, N, of the background. At the time of this experiment our100Fig. 5 The observed time (in the ordinate) for the point

vortex to climb the background vorticity hill iscompared with the theoretical prediction (in thediagnostic system was not sensitive enough to determine

the profile of the background vorticity. After theproposal of the theoretical model [10], we made aabscissa). From Ref. 5.

rough evaluation of the time scale by assuming that eachthe background distribution and N". It is found that theof the discrete vortices climbs the background hill

experimental values of ?s1p ar€ nearly the twice of Tyo6"t[5]. The difference in the constant factor is attributed tomerely replacing the logarithmic factor in Eq. (4) to 1 toindependently until they come close within the distanceof their diameter [4]. The naive evaluation gives a rightorder ofthe time scale, but reasonable scaling cannot be

avoid a mathematical singularity coming from theobtained in regard to the parameter dependence.
The difference in the applicability of the theoreticalmodel to the single-vortex and two-vortex cases has lead

simplification of the model.
The spiral structure observed in Fig. 3 rotates
differentially with respect to the point vortex. It can
produce time-varying electric field that may affect themotion of the point vortex. Oscillations have been
actually observed in the time history of the vortex
distance from the background center as shown in Fig. 4

us to consider that the background vorticity distribution
is significantly modified by the discrete vortices and thatit gives substantial influence back to the vortices. Figure
6 shows an example taken from recent data that cameavailable after a substantial improvement of the[5]. The oscillations are not expected from thediagnostics. It reveals that intricate structures appear in
the background distribution especially in the case where
the binary vortices are formed.

theoretical model.

From electron density distributions that areobtained with improved sensitivity and resolution of128

Kiwamoto Y. et al., Controlled Experiment on Two-Dimensional Vortex Dynamics in Electron plasma

FlnalState ls Slngle Vortex

lOpsec

Nv=l.3x107, N6=i.{v198

l830

60

10000

Hi'F--30.."mrnn"">
FlnalState ls Blnary Vortlees

l4psec

Ny=!.f1166, N6='l.l1l98

44130234

r 0000Ph.mr Lrb. Frc,lHg Kyoto U,
Fig. 6 Interacting two discrete vortices in a background vortex.

5. Holes and Humps in Background
Vorticity Distribution
Isolated depressions (holes) in the vorticity

14E912

g'og8

distribution have been invoked as a controlling factor in
the evolution of 2D turbulence [10,12]. Such holes havebeen observed in our experiments. But, in many cases,
we also have observed rings of dip or rings of hump ofoi56

6+92

Eo

the vorticity around unmerged discrete vortices. Holes

around discrete vortices have been invoked as aTime (ps)transport barrier of entropy in the statistical approach tothe formation of the vortex crystal [13]. Our observationshows that the circulation of the ring hole (decrementfrom the background) is 5 to 20 Vo of the clump in theo-l o.oe

core. Though the field of a discrete vortex may not be

fully shielded, the ring-shaped holes or sometimes ring0.01

Number of Back-Ground Electrons N

shaped humps appear to weaken the direct interaction
between the discrete vortices and to keep them awayfrom merging.
Our current understanding is that such contributionb (x107)of the background vorticity should be essential toFig. 7 (a) Two discrete vortices approaches faster as theaccelerating the process of organizing meta-equilibrium
structures and also effective to slow down the process ofcirculation of the background vortex f, * N,increases. Nb/108 = 0 (x),0.83 (A), 1.0 (t]), 1.6 (o),2.2 (.1. (b) Merging time ? scales as * N;k with kdisintegration until some imbalance grows so that
reorganization proceeds in a punctuated manner.

= 0.83 - 1. From Ref. 4.
129

Kiwamoto Y. et aI., Controlled Experiment on Two-Dimensional Vortex Dynamics in Electron Plasma(AIP1999) pp.99-105.

Detailed examinations about the generation mechanism

t4l Y. Kiwamoto, K. Ito, A. Sanpei, A. Mohri, T.
Yuyama and T. Michishita, J. Phys. Soc. Jpn.

and properties of the ring holes are reported in a
separate paper [8].

(Lett.) 68, 3766 (1999).

t5l Y. Kiwamoto, K. Ito, A. Sanpei and A. Mohri,

6. Conclusions

Phys. Rev. Lett. 85, 3173 (2000).

We have examined fundamental processes in theevolution of 2D turbulence under precise control of theinitial vorticity distribution. The plasma-physicalapproach is demonstrated to be effective in elucidating
intricate physical mechanisms that largely remain
behind the mathematical expressions of fluid dynamics,

L6l See, for example, J. Miller, P.B. Weichman and M.
C. Cross, Phys. Rev. A45,2328 (1992).
t7l K. Ito, Y. Kiwamoto and A. Sanpei, poster PII-52

in this conference.

t8l A. Sanpei, Y. Kiwamoto and K. Ito, poster PII-11
in this conference.

Acknowledgments

t9l K.S. Fine, A.C. Cass, W.G. Flynn and C.F.

The authors thank Prof. A. Mohri for stimulating
discussion. This work was supported by a Grant-in-Aid
from the Ministry of Education, Science, Sports andDriscoll, Phys. Rev. Lett.75,3277 (1995).
tl0l D.A. Schecter and D.H.E. Dubin, Phys. Rev. Lett.
83,2r9r (r999).
tlll T.B. Mitchell, C.F. Driscoll and K.S. Fine, Phys.
Rev. Lett. 71, I37l (1993).
|21X.-P. Huang, K.S. Fine and C.F. Driscoll, Phys.
Rev. Lett. 74,4424 (1995).
Il3)D.2. Jin and D.H.E. Dubin, Phys. Rev. Lett. 80,

Culture and partly by the collaborative research programof National Institute for Fusion Science.

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