J. Plasma Fusion Res. SERIES, Vol.4 (2001) 545-549

Vorticity Hole Surrounding a Point Vortex
SANPEI Akiox, KTWAMOTO yasuhitor and ITO Kiyokazu
Graduate School of Human and Environmental Studies, Kyoto IJniversity, Salcyo-ku, Kyoto 606-8501, JapantFaculty of Integrated Human Studies,

Kyoto University, Sakyo-ku, Kyoto 606-8501, Japan(Received: 5 December 2000 / Accepted,: 14 September 2001)

Abstract
Depleted regions (holes) in the vorticity distribution often play a controlling role in the evolution oftwo-dimensional turbulence. We report experimental investigations of the generation process and thegeometries of ring holes surrounding a strongly-peaked clump of vorticity (point vortex) immersed in a
background of the extended vorticity distribution.

Keywords:
electron plasma, vortex, 2-dimensional turbulence

1. Introduction

In this paper we examine the time scale of theformation process of the ring holes and characteristics ofIn the two-dimensional (2D) dynamics of anelectron column immersed in a strong uniform magnetic

field, the density n(x,y) and the potential distribution@(x,y) is equivalent to the vorticity ((x,t) and stream
function ry(xg of 2D Euler fluid, respectively. Areas of

their geometries in regard with the shape and strength ofthe extended vortex interaction with a point vortex orclump.
The experimental data are obtained with an electron

excessive (depleted) density in the 2D plane are calledplasma confined in a Malmberg trap. The details ofclumps (holes) in the vorticity distribution. In a nonneutral plasma, because of a lack of counter-charge
particles, coulomb force is partially shielded only bydensity depressions, i.e. formation of holes.
We have recognized the appearance of patchedholes or ring holes in the early phase of our experimental studies of point vortex motion in an extended
vorticity distribution. [1,2,3] Such ring holes have been

configuration and diagnostics have been reported
elsewhere. 11,2,3,71

2. Process of Hole Formation [8]Figure I illustrates the dynamics of a clump injected at I = 10 psec in a low vorticity region of an extended vortex. The darkness is proportional to thevorticity, and the black dot represents the clump. Thepeak vorticity (,s of clump is about 180 times that ofthe extended vorticity (6s, and its circulation l, is about
0.18 times the extended vortex' fr. The clump generates
a rotational flow around its center in the same direction
as that generated by the extended vortex (t = 20 psec).
The interacting flows make the clump climb along thegradient d(6Q)/dr of the background vortex trailing aspiral arm ofthe perturbed vorticity distribution. [3] The
clump drags a part of the low vorticity area into theintroduced in a theoretical model for explaining
crystallization of clumps in the perspective of statisticalmechanics. [4] In the model, a major role of the ringhole appears to be a barrier against local entropytransport. There is an experiment about formation of thehole by Huang [5], and Schecter theoretically atalyzed
the motion ofhole in an extended vortex. [6] It has been
pointed out that the holes have controlling effects on theevolution of turbulence because of their relatively slow
motion among clumps.
*Corresponding author's e-mail:@2001 by The Japan Society of Plasma
n50 I I 7 @ sakura.kudpc.lcyoto-u.ac.jp545

Science and Nuclear Fusion ResearchSanpei A. et aI., Vorticity Hole Surrounding a Point Vortex\q*(,E4{olo-oum5IN\uml#-ed--{qI12?b05101520
Fig. 1 Snapshots of the vortici

a ring hole around ar [mm]distribution generati ng
interacting with anFig. 2 Radial profiles of the extended vortex' The initial
location of the clump is indicated by an arrow.

extended vortex.

higher vorticity area of ex
spiral streak around its
differential flow around the cbetween the spiral streaks (rstreaks reconnect (t = 180d vortex and makes a(t = l0 psec). The

region involved in extended
that surrounds the clump (t =the ring hole has a long lifeti
the clump (r = 500 msec). We

icity forms a ring hole

generation process of the ri

shortens the distance

120 psec). The spiraland negative-vorticity

olGFmsec). Once generated
comparable with that ofine the speed of thehole in terms of thereconnection time Zp of the3. Empirical Scaling of

fv [m2lsec]

Time

Fig. 3 Reconnection time is plotted against 1". Eachsymbols corresponds to the different profiles int81

periphery of an exBy injecting a clump intended vortex that has variou levels and slightly differnt gradients of vorticity as llustrated in Fig. 2, weof a rins hole. The
have observed the formation
radial position of the injection indicated by an arrow.
We define Tp as the timeclump to the occurrence of
circulating around it. The
depends on fu and the local
background distribution (6(r)
Figure 3 plots Tp againstclump for four different dvortex. We have empiricallyholds between Z* and f" and t
B = 1(od(uldrl at the inirial

fitting lines for the dataFig. 2. The difference between closed and open
symbols is described in relation to Fig.s (a).

The coefficient connecting the empirical scaling isdetermined by plotting the observed Zp that is divided
by the right-hand-side against B. The final relation reads

from iniection of theion of the streaktime intricatelv

as,ition imposed by thet B^\n 1y^ltgotf| "rr*=ol#l\tr / \f"/the extended vortex.

circulation 1., of thetions of the extendednd that a oower lawt2(t)where a = 5.2 x 102 psec. Figure 4 plots the observed
values of the reconnection time Tp-B;" against theassociated values Zp-4a1 obtained from the empirical
relation (l). It demonstrates that Eq. (1) represents thewide class ofthe observed data quite satisfactorily.
A recent paper by Durkin et al. l9l has proposedthat the wave-braking time T.6 e /'t1n1-r is a good

the power depends onof the clump. Thein Fig. 3 provide anempirical relation, Zp * (fe/f $ilbz , where the fitting-r, andparameters are lo = 15
Fo = 8.9 x lgs

measure of describing the reconnection time, where .l =m-lsec-2.

546Sanpei A. et al., Yorticity Hole Surrounding a point Vortex(a)103.n=CLxEPIJJ

F

100 z
102

103

Tn ca [ps](b)Fig. 4 The observed reconnection time is plotted against
the prediction of the empirical scaling (1).

fulf5. We have examined the correlation between Zp
obtained in our experiments and the observed i, but did

not obtain meaningful relation between them. Ourcurrent evaluation of this model is that it cannot beapplied to the general cases with inhomogeneousbackground distribution though it may bevalid to a
special case of flat disk distribution of the extended

xvortex.

Fig.5 Unified functional dependence of the recon-So far we have not obtained a full physical explanation of the relation (1). Noticing experimental data
indicating a functional relation between B and the shear
A = -rddlo?)ld.r that has global meaning, we expect thatthe relation will be reconstructed from a shear flow
model including the contribution of the clump.

nection time on yand B is examined in the form ofc =flxl. The functional form differs depending
whether y> 0 (a) or y< 0 (b).

TRf(8")

4. Functional Analysis of Examination of
Empirical Relation

The functional form of/(x) depends on the sign of
7 Figure 5 plots the function/(r) for 7> 0 in (a) and for
Jz< 0 in (b) in solid curve. The experimental data areplotted also in corresponding graphs. Here x and t are

Now we analyze what effect the profile ofextended
vortex has on the reconnection time. Let us set ]z =ln(f,/To) and p = BlB6. Then the right hand side of Eq.
(1) is written as

f Gl= ap-'o expl-y/B'1.

determined from the observed parameters. We canobserve an excellent agreement between individual data
and the unique function /1x; that has two branches ofform appearing from a degenerated expression in Fig. 3.
The data points with y> O and x < I are represented
by closed symbols, while other data points are given by
open symbols. This classification traces back to Fig. 3(2)

We renormali ze Eq. (2) by dividing it by its value at p,= 2lYlt'' to obtain(B)(r - vx,ll.
"j',l:ff(8")=*-,,."0[rL^l4lvl'(4)and Fig. 4, and the separation point.x = 1 is indicated by
the symbols x on the fitting lines in Fig. 3.

(3)

I

When 7< 0, clump is weak and it causes a small
perturbation in the extended vortex. This region

Here we have introduced a new variable I =plB". focompare the observed reconnection time Z* with Eq.

therefore may be classified as the weak-point-vortex
region. When 7> 0 so that the strength of the clump is(3), we define the normalized reconnection time as547

Sanpei A. et dl., Vorticity Hole Surrounding a Point Vortex

regions separated atftln(f,/T6) so that thended vortex. On theother hand, in the region of x < 1. the effect of theion process as may beclump dominates in theunderstanding, the experimental data in Fig. 6 clearly
indicate that fg lies from 5Vo to 20Vo of 1,.
This experimental evidence might raise a question
to the fully-shielded vortex model in Ref. [4] and to anaive assumption of fs = lu introduced in Ref. [9]. Heresuggested by the locations of

let us note that the vortex-merging is not driven directly

not weak,/(r) is separated into.r = 1.In the region of x> l, B
dynamics is dominated by thesymbols in Fig. 3.

by the force between symmetric vortices but thatmerging require asymmetric deformations in the5. Geometry of Ring

distribution of the vortices. [10] The merging drive insuch a perturbational order may be effectively modified
by some redistribution in the hole vorticity (6(.r,y) even

Now we report about the si of the ring holes. Thering hole depends on
radius Rse of the outer edge offu and profile (6(r) of the e tended vortex. As I."of the radius of theincreases, RHe increases to thestagnation zone I = (l,l2nlAl)tt2
The maximum depth in ( (.r,y) of the ring holealso depends on l" and (6(r).

depth ("(.x,y) standsfor the decrement from therbed level of theif fH << f,.

6. Conclusions
We have studied the generation process andgeometrical properties of ring holes around a point
vortex or clump. We have obtained an empirical scalingthat includes several parameters relevant to the vortex

indicate that
extended vorticity (6Q). Ourof the local (6(r), andthe maximum (s is less than
(66 of the clump. We
less than l%o of the peak vortici
ler than the decrement
should note that 6H is muchin the wake of ain the open spiral streak
clump that can reach almost
0OVo of (6. [31Let us examine shieldinethat the rine holeion of the hole, fg = J
may have by defining the ciof ls to f, againstdxdy(s@,y). Figure 6 plots thel/fR-cAL. Here we choose Z a1 in (1) because it
vant to the build-up ofincludes several parametersof the strength to
the ring holes and could be agenerate the holes. Though aive correlation may beexpected between ls/|" and I [-44p, the observations

dynamics to predict the reconnection time of a spiralstreak. The geometrical properties of the ring holeindicate that the ring holes do not completely shield theclump's field. Our observations may request
reconsiderations about the dynamics of interacting
clumps surrounded by modified background vorticity
distributions.

Acknowledgments
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 andCulture and by the collaborative research program ofWe don't have alead to somewhat puzzling reclear physical meaning of thi correlation, and need
all this state of ourmore consideration about it.

National Institute for Fusion Science.

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