J. Plasma
J.
Plasma Fusion
Fusion Res.
Res. SERIES,SERIES,Vol.
Vol.6 6(2004)(2004)125–129
000–000

Critical Contribution of Low-Density Background Particles inthe Formation of Ordered Structures in Pure Electron Plasma
KIWAMOTO Yasuhito, SANPEI Akio, SOGA Yukihiro, AOKI Jun, YATSUYANAGI Yuichi
and OZAWA Tatsuma
Graduate School of Human and Environmental Studies, Kyoto University, Kyoto 606, Japan(Received: 10 December 2003 / Accepted: 11 March 2004)

Abstract
We study how low-density (background = BG) electrons affect the generation processes of ordered structures inthe distribution of high-density electrons (clumps) in two-dimensional dynamics transverse to a homogeneous magnetic
field. Here, by reviewing recently-published work [1,2], we discuss mainly on two experimental results: (1) twoclump dynamics focusing on bifurcation between merger and close-separation under critical contribution of BG
electrons, and (2) their contribution to advecting three clumps to a symmetric configuration and sustaining the geometryof an equilateral triangle that represents a unit cell of vortex crystals. We regard the two phenomena are among key
processes supporting the self-organization of ordered structures in many-particle systems with long-range interacting
force.
Keywords:
nonneutral plasma, vortex dynamic, self-organization

1. Introduction

with pure electron plasmas is the discovery of crystallizedstates of vortex strings. The first experimental study wasinitiated by generating many string distributions of electrons
in the nonlinear stage of Kelvin-Helmholtz instability. The
subsequent process of the relaxation of the vortex strings ischaracterized by repeated organization of highly symmetric
arrays of strings whose number decreases intermittently as afunction of time. [7]To explore abundant physical significance involved in the
self-organization process, we have carried out a series ofexperimental investigations by singling out a simplified
configuration in the process. [1,2,8,9,10] Our typicalconfiguration consists of a set of discrete distributions ofelectrons (clumps) initially immersed in a well-definedcontinuous distribution of tenuous electrons that act as acollection of background vortices (BGV).
The first topic is focused on experimental investigations
on how the merger of two clumps is accelerated or blocked
by different levels of BGV to a degree substantially different
from the clumps’ dynamics in vacuum. [1,8] Modificationsin the BGV distribution generated by the clumps’ motion exert
critical effects on the merger or generation of a closely tiedstate of the clumps. The second topic is on how the unit cellof vortex crystals is established in interaction with otherclumps and fragmentary vorticity distributions consisting ofremnants of preceding vortical interactions. [2] Here a set ofVortex has become one of the key concepts incontemporary physical science. [3] Quantized vortex
represents major property governing the dynamics of superfluid. [4] The accumulation and fusion-triggered abrupt
release of the vortex strings in the super fluid core is called
for as a possible mechanism of glitch in the spinning speedof a pulsar. [5] Most prominent features of Bose-Einsteincondensates include vortex dynamics of probability
distribution functions. [6] Needless to say, vortex dynamics
appears in fusion-oriented plasmas in many essential ways
including convective cells that enhance unfavorable radialtransport and internal transport barrier that benefits theconfinement. There is a possibility that one can explore thefrontier of plasma physics as an active and easy-to-experimentfield to investigate vortex as a concept common in the wide
spectrum of physical science.
Strongly magnetized pure electron plasmas arecharacterized by the long-range coulomb interaction without
effective shielding and by incompressible two-dimensional(2D) dynamics equivalent to the 2D Euler fluid. The particle
density n(x,y) is related to the vorticity ζ(x,y) by ζ = en/ε0 B0 ,and the circulation Γ is defined as Γ = ∫ dxdyζ(x,y) = eN/ε0 B0 L. Here N is the total number of electrons trapped within
the axial length of L in a homogeneous magnetic field B0 .
One of the culmination among studies of fluid dynamics

Corresponding author’s e-mail: m52279@sakura.kudpc.kyoto-u.ac.jp1125©2004 by The Japan Society of Plasma
Science and Nuclear Fusion ResearchKiwamoto Y. et al., Critical Contribution of Low-Density Background Particles in the Formation of Ordered Structures in Pure Electron PlasmaFig. 1

Snapshots describing clumps’ dynamics in BGVD with Γb ∝ Nb /108 = 3.55 (a) and Γb ∝ Nb /108 = 3.39 (b). Two clumps with equal
circulation Γc ∝ Nc /107 = 0.6 are placed symmetrically in the initial BGVD. The luminosity nonlinearly increases with electron
density and saturate at high level. The panel size is 37.4 mm × 37.4 mm at the mid-plane of the trap.

three clumps with different number Nc ∝ Γc of electrons areobserved to settle down to the vertices of an equilateraltriangle in the presence of BGV, roughly with a speed
increasing with Γb ∝ Nb , the circulation of BGV.

2. Bifurcation: merger or close separationof two clumps [1]Stable maintenance of relative locations of clumps is one
essential feature of vortex crystals, and merger between themis another feature associated with the transition of the vortex
configuration to other with decreasing number of clumps. In
the relaxation processes involving the crystallization, the BGV
distribution (BGVD) is generally low, and contributions fromthird clumps add to the BGVD’s contribution. Here we
examine the interaction between two clumps by smearing
other clumps into BGVD.
Two clumps in vacuum are known to merge when placed
within the distance of 1.6 × vortex diameter. [11] If a tenuous
distribution of electrons (∝ BGVD) fills the space around theclumps well-separated in space, they approach quickly as
shown in Fig. 1 in the time scale τV ≈ 50 µs of vortex
orbitation. [1] If BGVD is high and peaking enough, theclumps continue to approach and merge in the time scale ofτV . [8] However, if the BGVD is low and / or rather flat inshape, the clumps rebound at t ≈ τV and stay apart for period
longer than τV by orders of magnitude. The conditions
relevant to the formation of crystals are positioned in a narrow
parameter range between the two cases.
We examine the separation between the two clumps as afunction of time for slightly different initial distributions of
BGV as shown in Fig. 2(a), that are systematically varied witha precise electronical control. The ratio between the peakdensity of the clumps and the local density of the BGV istypically nc /n b = ζ c /ζ b = 13–15. We have measured thedistance D(1 ms) between the clumps at 1ms for various
initial conditions, and extensively examined its correlation
with various parameters characterizing the initial BGVD. The

Fig. 2(a) Radial profiles of the electron density distributionnb (r ) = ε0 B0 ζb (r )/e constituting BGVD with nb /108 =3.55 (●), 3.46 (closed diamond), 3.25 (closed square),3.14 (×), 3.39 (䡬), 3.08 (square), 3.13 (open diamond),
3.17 ( ), and 3.16 ( ). The arrow indicates the initialradial locations of the clumps. (b) The inter-clump
distance at 1 ms is plotted against ∇nb /nb2 as evaluatedat the initial locations of the clumps.

time 1 ms, evaluated as the geometric mean of 2τV and thecoulomb collision time ≈ 10 ms, stands for the time domaincharacterizing the asymptotic state of the vortex dynamics
before collisional dissipations set in. Clear correlations are2126

Kiwamoto Y. et al., Critical Contribution of Low-Density Background Particles in the Formation of Ordered Structures in Pure Electron Plasma

not obtained with most of the initial parameters or their
combinations except between D(1 ms) and ∇ζ b /ζ b2 ∝ ∇nb /nb2
defined at the initial position of the clumps. The results ofthe analyses are summarized in Fig. 2(b). The smooth fitting
curve, however, gives two values of D(1 ms) for a given valueof ∇nb /nb2.
The absence of other relationship of D(1 ms) with initial
parameters that resolves the degeneracy have led us toexamining noticeable structures in the vorticity distributions
during the evolutionary phase of the clumps’ dynamics. Theclumps are observed to stagnate at 50–100 µs before merger
or rebounding. The BGVD’s at this moment are found toexhibit systematically varying pattern of belts of depleted
vorticity around the clumps. Compare the panels at 50 µs in
Fig. 1. Systematic comparison of the location and length of
the belts with the asymptotic separation of the clumps has
led us to a model that evolution of this structure in BGVD
prevents the merger.

For quantitative description of this model, we evaluate
the depth ∆nb of the depletion and the sum ∆r of the width of
the belt along the chord connecting the peaks of the clumps.
Figure 3 plots D(1 ms) as a function of ∆nb ∆r. We see that
all the data points in Fig. 2(b) are continuously displayed
along the linear fitting line in Fig. 3 The resolution of thedegeneracy indicates that the formation of ordered structuresof clumps critically depends on the fluctuating structures ofthe ambient BGV that are generated in mutual interaction withthe clumps and with other parts of BGVD.

3. Formation of unit cell of vortex crystal[2]
We add one more clump to the previous study in order
to examine the process of the formation of a unit cell of 2D
symmetric arrays of vortex strings. Fourth clumps and BGV
are smeared into a broad profile of the average BGV
distribution. Figure 4 shows the dynamics of three clumps
without BGVD (a) and with BGVD (b) starting with equal
circulation Γc ∝ Nc from a linearly aligned configuration. TheBGVD curbs the excursion of the clumps and adjusts theirrelative positions to the vertices of a triangle with equal sidelengths in apparent contrast with the motion of the clumps
without BGVD that do not settle down to any stationary
configuration.
This positive contribution of BGVD is also observed forthree clumps with different circulations that finally form anequilateral triangle configuration. An appreciable difference
is observed in the pattern of BGVD around each clump: Thestronger clump (with larger circulation) has wider range of adepressed distribution of BGVD around itself. Ourquantitative analysis has shown that the negative circulation– Γ h of the depression, as measured with respect to theunperturbed height of BGVD, increases with the circulation

Fig. 3The inter-clump distance D(1 ms) is plotted against∆nb ∆r at 50 µs starting from different distributions ofinitial BGV. The inset illustrates the definition of ∆nband ∆r.

Fig. 4

Snapshots of the clumps’ dynamics in vacuum (a) and in BGVD with Γb ∝ Nb /107 = 9.4 (b). Three clumps with equal strength ofΓc ∝ Nc /107 = 0.74 are placed in straight line as the initial condition.

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Kiwamoto Y. et al., Critical Contribution of Low-Density Background Particles in the Formation of Ordered Structures in Pure Electron Plasma

Γc of the surrounded clump but does not exceed 20%, i.e. Γh< 0.2 Γc .
The degree of the contribution of BGVD in the formationof an equilateral triangle is most apparently observed in termsof the total circulation Γb ∝ Nb of BGVD. Figure 5 illustrates
this feature. Here three equal clumps are placed symmetricallyin BGVD at the locations indicated with arrows as illustrated
in the inset. The height and profile of the initial distributionof BGV are varied systematically. The abscissa scales theradius from the axis to 2 cm, while the ordinate scales theBG density up to 2 × 1012 m-3.
The main body of Fig. 5 plots the degree of proximity
to the equilateral triangle of the three-clump configuration interms of the symmetry parameter S = 12√⎯3A/l2, where A isthe area of the triangle with total side-length of l. S is
maximized at 1 when the triangle is equilateral. Figure 5indicates that, after repeated variations in shape between thinand fat triangles in the time-scale of ≈ ms, the cell of theclumps asymptotically approaches to the symmetric form with
S = 1. The asymptotic dependence is observed to beapproximately 1 – S ∝ t-1.
The time τ S required to settle closely down to theequilateral cell is most strongly correlated to the circulation
Γb of BGVD, and τS decreases with Γb . Because the clumps
and BGVD rotates 10–100 times before settling, differencesin the profile of initial BGVD may not dominate the evolution
path of the system. On the other hand the rugged distributionin BGVD generated by the interaction with clumps may be
more influential as we have found in the previous section.

with a narrow density range above the upper bound of thethree-clump experiment. The assisting contribution of BGVD
in the symmetrization of the three-clump cell is effective atdensities roughly as low as 1/60 of the case where thebifurcation between merger and binary vortex is observed. We
also have noticed the tendency that the size of the equilateraltriangle of the clumps decreases as ζ b or Γb increases and theprobability of clumps’ merger goes up to a statisticallyobservable level. Our current understanding is briefly
summarized that the merger process between clumps is
associated with a large gradient in the local vorticity
distribution that can be generated either in BGVD or by otherclumps and that the merger is hindered if the sign of the local
gradient is reversed.
Next we extend our discussion to microscopic structuresin BGVD that induce fluctuations in the velocity of theclumps and try to relate them to the geometrically determinedparameter S. The speed of random motion δv of each clump
can be evaluated from density-weighted average of the E × B
drift velocity of electrons constituting the clump. [12,13] The
open circles in Fig. 6 are obtained as the deviations from therigid-rotating velocity of the triangle. Schecter et al. take thisscheme in the analysis of many-clump system. [14] Whenapplied to the three-clump system, δv is larger than expectedfrom the almost stationary shape of the triangle over 10 ms
as observed in the experiments.
Further analysis indicates that a differential rotation
persists in our experimental configuration with fewer clumps.
If we admit, based on our experimental observations, thatthree clumps need not be stationary in the frame of the whole
vortex system and that they can roll in the BGVD with arotational shear, the random velocities are evaluated low asplotted with solid triangles in Fig. 6. This analysis suggeststhat the clumps move coherently among themselves keeping

4. Discussion and conclusion

1-S

We briefly compare the results of the two-clump andthree-clump experiments. As read in the caption of Figs. 1-2,4-5, the two-clump dynamics has been examined in the BGVµTime (µs)
Fig. 5

Deviation 1 – S from the symmetric configuration isplotted as a function of time. Each symbol corresponds
to different initial distribution of BGV as shown in theinset. The electron number constituting each profile isNb /107 = 0 (×), 0.25 ( ), 0.52 ( ), 9.4 (closed diamond),
21.0 (square), 24.8 (䡬). Three clumps with equal
strength of Γc ∝ Nc /107 = 0.74 are placed in straight line
as the initial condition. Their radial locations are on theaxis and at symmetric points as indicated by arrows.

Fig. 64128

Clumps’ random (thermal) velocities are plotted against
1 – S. Solid triangles represent mean random velocities
causing deformation from the instantaneous triangle(not necessarily equilateral) that rolls around in BGVD.
Open circles stand for the mean velocity causing
deviations from the instantaneous clump configurationin the global coordinates that rotate rigidly.

Kiwamoto Y. et al., Critical Contribution of Low-Density Background Particles in the Formation of Ordered Structures in Pure Electron Plasma

the configuration of an equilateral triangle. As far as we
confine ourselves to the observed data, the numerical analysesof the flow field provide self-consistent view of the clumps’dynamics. However, the physical mechanism that supports thepreservation of the symmetry of the triangle moving in theBGVD has not been definitely identified.
It may be of worth to take Aref’s theoretical model intoconsideration. [15] The Aref model tells that three clumps,
once placed at the vertices of an equilateral triangle, remains
there in interaction with other clumps even when thecirculations are unequal. Experimentally we have observedthat this configuration is unstable most probably due tointeraction with image charges induced on the boundary wall.
Destabilizing contribution of the wall has been theoreticallypointed out for clumps with equal circulations. [16]Experimentally observed dynamics appears to be moreunstable than theoretically predicted. Our observations
reported here indicate that, with all these destabilizing factors,
the BGVD works positively to generate symmetricconfigurations by transporting the clumps to symmetricpositions, by shielding destabilizing wall effects and byremoving random velocities.

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