J. Plasma Fusion Res. SERIES, Vol.4 (2001) ll7-123

Experiments and Theory on 2D Electron Vortex Dynamics
in Sheared Flows
KABANTSEV Andrey*, DRISCOLL c. Fred, DUBIN Daniel and SCHECTER Davidl
Physics Department, university of califurnia at san Diego, La JoUa, califurnia 92093
1

National center for Atmospheric Research, Boulder, Colorado 80307(Received: 5 December 2000 / Accepted: 16 August 2001)

Abstract
The motion of self-trapped electron vortices on a background vorticity gradient are studied
experimentally, and compared to numerical simulations and analytical theory for 2-dimensional
incompressible, inviscid flows. These vortices can be either clumps (vorticity excesses) or holes (vorticity
deficits). The vortices mix the background vorticity and tend to level the background gradient. As aresult, conservation of momentum dictates that clumps move up the background gradient, whereas holes
move down the gradient. The rate of this drift motion is dramatically different for prograde (rotating withthe background shear) versus retrograde vortices: a prograde vortex moves at a much slower rate than aretrograde vortex, due to rapid nonlinear mixing of the flow field. We have measured the clump and hole

velocities for a wide range of vortex strengths, background gradients, and background shear rates. The
measurements are in quantitative agreement with theoretical predictions and numerical simulations.
When parameteized,by the dimensionless vortex strength (, lru (where I is the size of the trapped region
around the vortex, and ru is the vortex orbital position, representing a characteristic size of the
background streamlines). Experiments also show that the rapidly moving retrograde vortex leaves a spiraldensity wake, and that instability of this wake eventually generates many long-lived holes, which
contribute to the late-time "turbulent" noise.

Keywords:
non-neutral plasma, 2-dimensional flow, rotating fluid, monopolar vortex, vortibity gradient

1. Introduction

which causes the bulk kinetic energy of the flow toslowly dissipate. However, 2D flows in plasmas,

Jupiter's Great Red Spot, hurricanes and ocean eddies
are just some of intriguing examples. A monopolar
vortex consists of a patch of fluid rotating around its
center, and self-trapped implies the background shear
does not destroy it. Generally, both mutual advection
and the interaction of vortices with the background
vorticity gradient play important roles in 2D hydro-

atmospheres and oceans can rapidly relax to metastable

dynamics, including various aspects of relaxation toward

states before viscosity affects the dynamics.

an ordered state and self-organization of turbulence. In
some cases, turbulence relaxes through the migration of

The free relaxation of two-dimensional (2D) turbulence has been actively studied for decades, withmany applications in astrophysics, geophysics andplasma physics. One relaxation mechanism is viscosity,

Self-trapped 2D monopolar vortices, immersed in awide background vorticity distribution, are quite
common phenomena in nature and experimenrs;*

vortices to extrema in the background vorticity. Thus,@2001 by The Japan Society of Plasma
Science and Nuclear Fusion Research

Corresponding author's e-mail: akabantsev@ ucsd.edu

ll7Kabantsev A. et aI., Experiments and Theory on 2D Electron Vortex Dynamics in Sheared Flows

the vortex/background interaction can cause the vortices

to move transverse to the direction of the backgroundflow, either up or down the vorticity gradient. Thisapplies both to magnetized non-neutral plasmas [1-4]and to atmospheric and oceanic turbulence and eddies[5-8]; on geophysical scales, the latitudinal variation of
the Coriolis acceleration gives a rotational shear in the

(a)

(b)

Fig. 1 Schematic representation of an axisymmetricbackground with either clump (a) or hole (b) on it.
Local mixing of the background increases (l)0. By

background (potential) vorticity gradient, called the
B-effect.
The prediction of hurricane tracks is a problem ofgreat practical importance, and a lot of theoretical work
has been devoted to the subject. Schecter and Dubin
recently performed a theoretical analysis of this problem

conservation of Pr. clumps and holes react
oppositely.

vorticity. In shear flows, in addition, they have to bein the specific case when pointlike monopolar vortex ofrelatively weak strength is placed within a diffuse,circular vortex (background vorticity) [9]. Relatively
weak vortex strength means that the size of trappingclassified as either prograde or retrograde. A prograde
vortex rotates with the local background shear, whilerotation in the opposite sense defines retrograde
vortices. In presented on Fig. I background a clump isregion in a background flow is much less than aretrograde and a hole is prograde.
We emphasize here that the conventional terms of
cyclones and anticyclones are not adequate to the
problem because they can be either clumps or holes,

characteristic size of its streamlines. The analysis was

focused on cylindrical shear flow, where the flow
conserves canonical angular momentum. Conservationof momentum then dictates that positive vortices

depending on which hemisphere they are located in.
Also, prograde or retrograde refers not to the background rotation, but to the shear in the background("clumps") move up the gradient, whereas negative
vortices ("holes") move down the gradient. Generally,
the vortex speed is proportional to the background
vorticity gradient; however, a vortex that is progradewith respect to the background shear moves about 20
times slower than a retrograde vortex of equal strength.
Separate theories were given for the motion of prograde
and retrograde vortices. Both theories compare
favorably to vortex-in-cell (VIC) simulations and to therotation.

The opposite drift of clumps and holes can beeasily explained by momentum conservation. Similararguments have been used to explain the motion ofphase-space density clumps and holes in plasma
turbulence [10]. We focus on cylindrical geometry,
where the flow conserves canonical angular momentum'

experiment.

Pr: I d2r (r2 .

It is convenient to write Pe in terms of a back-2. Simple Galculations of Gradient-Driven
Driftground contribution and a vortex contribution,

We focus on specific case where initial background

vorticity ({r,0) is positive, axisymmetric andPr=monotonically decreasing with radius at t = O (Fig. l).
governed by the 2D Euler equations:d(Y(=0, i=ixYV, Y'V=e .

((r,o,t1=fo(r')o + f"r] .

A)

Here 15 > 0 is total circulation of the background flow,
I, is the vortex circulation, r" is the radial position of
the vortex and (r2)6 denotes the (s-weighted spatial
average of 12. As shown in Fig. 1(a, b), both clumps andholes partially mix the background vorticity and act to

We neglect viscosity and consider flows that are-+v
At

I d'rr2(1)

level it. As the background is leveled, (r?)o increases,
since d(6ldr < 0. To conserve Pe, the clump (fn > 0)
must decrease ru and climb the background gradient'whereas the hole (f" < 0) must increase ru and descend

tlere,i(r,0,t) is the velocity tield., ((r,O,t) = ? ' V ,. J it
vorticity, and ty(r,0,t) is a stream function. The
coordinates (r,0) denote polar radius and angle, and t is
time. For analysis, the vorticity is decomposed into
vortex (v) and background (b): ( = (o + h.
As we used it before, the vortices can be classified
as either clumps or holes, depending of the sign of their

the gradient.

When there is no local vorticity gradient, local

mixing does not affect the background vorticity
118

Kabantsev A. et al., Experiments and rheory on 2D Electron vortex Dynamics in sheared

distribution. Therefore, where d(,oldr = 0, there is no
local mechanism for the vortex to exchange angularmomentum or linear momentum with the background.
This suggest that clumps will settle on hills of the
background vorticity, and that holes will settle in

Flows

retr

troughs, where d(6lAr = 0.

o)

We now determine the radial speed of the vortex.
Single monopolar vortex consists of a patch of fluid

Fig.2 A sketch of initial streamlines in the vicinity of a
retrograde clump (a) and a prograde hole (b) in arotating around its common center. The vortex's
dominant translational motion is rotation around thecenter of the background. We work in this rotating

shear flow C)6 < 0. The center of a circular
background flow locates far above the drawings.

frame, so the vortex is nearly stationary, and we defined
a local coordinate system centered at the vortex. Figure2(a, b) shows the initial streamlines in the vicinity of atrapping length l,, which serves as a small length-scale
cut-off. Thus, the linear theory breaks down for times
greater than the orbital period of a fluid particle locatedinitially at distance > (. from the vortex center. For a
retrograde vortex [Fig. 2(a)), the fluid particle is in the

retrograde clump (a) and a prograde hole (b). The
stagnation points in Fig. 2(a) are at distance l, aboveand below the clump, where(:l, /2nr,dlloe,)outer trapping region, and its angular velocity R(| (. (inthe rotating frame) tends to zero for small vortex(3)

circulation; whereas the length ofthe trapped orbit tends
to a finite vallue of 4nr,. Thus, for retrograde vortex, theorbital period diverges to infinity as the vortex strength
goes to zero. For prograde vortex [Fig. 2(b)], the fluid
particle has an orbit of length of 2tc{, and orbital velocity

is the trapping length, which depends on the clump
circulation and on the background shear (here, e6(r) is
the initial rotation frequency of the background).

We treat the vortex and the disturbance that it
generates as perturbations to the initial axisymmetricshear flow (o(r)

that is also proportional to l; therefore the orbital period= (,"(r, t = 0). In case of a weak vorrex,

remains constant as the vortex strength goes down.

with trapping length much less than the streamline size,

Thus, the time scale for the linear theory to fail becomes

i.e. (,lru << 1, it is possible to linearize the Euler
equations tEqs. (l)l to obtain the evolution of the
background vorticity perturbation. It is this background
perturbation that causes the vortex to leave its initial"instantaneous" relative to the time scale of vortex
motion.

However, a workable estimate of the radial drift
velocity of prograde vortex can be obtained using

circular orbit.
Using an unperturbed orbit approximation, after astandard (but lengthy) calculations, Schecter and Dubin
have shown [9] that for sufficiently weak vortices thetime asymptotic radial drift of the vortex center issimple "mix-and-move" analysis; here the prograde
vortex substantially flattens the mixing layer, and moves

a distance determined by conservation of angularmomentum during the time needed to flatten the mixing
layer. Suppose that the vortex levels the entire mixing
layer and has a negligible effect on fluid outside theapproximately given by!'= xthl'tn1c.r"/().

mixing layer. This mixing increases the backgroundcomponent of Psby an amount(4)

LPe.a=-an('orl13

Here, '+/-' is for clumps/holes, and c is a factor of order
1, determined by ru and the form of (s(r).

(s)

By conservation of Pe, the hole must increase ru by an

However, by comparing this analytical prediction
with results of VIC simulations [9], Schecter and Dubin
have found that the linear equations of motion apply
only to retrograde vortices, and that nonlinear effects
must be kept to explain the slower drift of prograde

amount Ar. Assuming that Lrlr, < l, we haveo,:+=(*.

(6)

To obtain the hole velocity, we also require an

vortices.

estimate of the time Ar for the mixing layer to flatten.
Presumably, this time is given by the orbital period of a

Linear theory breaks down when the trapped fluid
becomes mixed over a length scale greater than thefluid element near the separatrix that encloses thell9Kabantsev A. et aI., Experiments and Theory on 2D Electron vortex Dynamics in Sheared

mixing layer. Since the angular speed of this fluid
element is lC)6l I (in the rotating frame), and the orbit
covers 4E radians (2zr in both directions), we have

4|t

Lt =-.

Q)

lnilr

bounce motion of individual electrons in comparison
with the column bulk rotation time (- 50 ps) effectively
averages over any z variations. Together with small
electron gyro-radius (- 10 pm), it allows us to use the2D E x B drift approximation [11,12] of the system
macroscopic dynamics

Finally, the velocity of the prograde vortex is given bydr,- Lr = _L tr (r.

dt Lt

4x

Flows

4dt *i. Vn=0, i =2xcYrpl B, Y'rp=4nen. (9)(s)

which is isomorphic to the 2D Euler equations [Eqs.
(l)1.

Note that the l-scaling in linear theory [Eq. (4)]Comparing the drift-Poisson equations (9) to theEuler equations (l), we see that the vorticity ((r'q't) is
directly proportional to the electron density n(r,4't), i-e.
ty(r,9't) is directly€ = 4necnlB; and the stream function

differs from the l-scaling in Eq. (8) by a factor ofln(c.r"ll). Therefore, our estimate suggests that a
retrograde vortex, which foltows the linear theory, will
move much faster than a prograde vortex for (' lr, < I'
On other hand, when I /r, - l, the ratio between theradial velocities of a retrograde vortex and a prograde

proportional to the electrostatic potential E(r,0,t), i-e. ty= calBt while nonzero dtPldr at the wall corresponds tothe free-sleep boundary conditions. Therefore' thevortex will be about 2tt2.

vorticity is simply advected in the incompressible flow
with electron density, the streamlines conform to theequipotential surfaces, and vorticity measurements are

3. 2D Fluid Experiments with MagnetizedElectron Plasmas

equivalent to density measurements.
In the experiment, we first inject and trap a stablequiescent column with monotonically decreasing andMany features of turbulent relaxation processes in
2D Euler fluids have been studied using electron
plasmas confined in a Penning-Malmberg traps [1-4].
Conventional fluid experiments are difficult tomanipulate and diagnose, and are subject to undesired
viscous and boundary effects. With low viscosity,
circular free-slip boundary conditions, simple
manipulation technique, and accurate diagnostics,

cylindrically symmetric initial density profile ns(r).

Then, we either create a narrow hole of radius pu < Roat the center of electron column by temporarily
decreasing the confinement voltage at one of the end
cylinders and allowing electrons to escape from the trap;
or combine the column with a small peripheral clump

magnetized electron columns provide excellent(which is prepared in adjacent trap). The column witheither a clump or a hole then evolves for time t, aftet
which the electron column is dumped axially onto aphosphor screen. The 2D density image n(r,O,t) isrecorded with a CCD camera, and €(r,0,t) is obtained

opportunities to study 2D turbulence. However, few
experimental studies have quantitatively measured 2Dvortex motion in a shearing background flow. So far,laboratory observations have been reported by Huang etal. t21, who measured only a single datum of the drift

velocity of two holes down a vorticity gradient; andmore recently by Kiwamoto et al. [4], who' on thecontrary, only examined the dynamic behavior ofelectron clumps immersed in a background vorticity.
The experimental apparatus used at UCSD [2'3] isshown in Fig. 3. Electrons from a tungsten filament arecontained inside a series of conducting cylinders ofradius R* = 3.5 cm enclosed in a vacuum chamber (P 10-10 torr) with uniform axial magnetic field B = 4 kG.
The magnetic field provides radial confinement, andnegative voltages (% = -80 V) applied on the two end

Sr,4l}

cylinders provide axial confinement. A trapped electron
column typically has maximum density ne - 107 cm-3,
radius Ro = 2 cm, axial length Lp = 35 cm, and electron
thermal energy T" = | eY - The rapid (- 1 ,tis) axial

Fig. 3 Experimental apparatus (Penning-Malmberg trap)
for 2D fluid experiments with magnetized electron
plasmas.

r20

Kabantsev A. et al., Experiments and Theory on 2D Electron Vortex Dynamics in Sheared Flows

directly. Although this imaging technique is destructive,
the shot-to-shot variations for nearly identical initialconditions are small (< l%o in azimuthally averaged
local density), so that the time evolution of a flow can

2.5

be studied.

1.5

2Figure 4 shows a typical example of background
vorticity (s(r) and rotation frequency C)s(r) profiles, aswell as a profile. The chosen background vorticity
distribution represents a large class, where plasma
density decreases radially outward, and the radialderivatives of 9-averaged (6(r) and Q6(r) vary slowly
within r < Ro. Their characteristic values, used in thecurrent experiments, were in the bands 50 < 6d < 500
msec-r cm-r and 10 < C)6 < 50 msec-l cm-l, so that thecondition of non-critical shear strength I9l 6,i/O6 > 1 isRn$]

{ #ifsl}
fl{xlffs')'"

K'

0.5

a 0.5 1t.5 2 2"5 3 a5 4
r (cm)
Fig. 4 Graph showing the radial cross-sectional

distribution of the vorticity (o of a typicalbackground flow, its rotation frequency f,)., aswell as radial position and the vorticity of aalways satisfied.
We can easily create clump (or hole) characterized

generated clump (",u.0 (dashed line).

by approximately "gaussian" density (vorticity)
distribution with the maximum density zu comparable tothe background density scale n6, and the characteristic

radial extent p, > 0. lRp = 0.2 cm. In this case, theaccessible region of vortex strengths is p"/Ro < | lr, <(Rrlp,)t'', i.e., 0.1 3 (,1r,3 3. Thus, we can compare ourexperimental results to the linear analysis of vortex
motion for (, /r" < 1, and also obtain results where thetheory is invalid at (,lrn - l.
Figure 5 shows the evolution of the measured
vorticity distribution when either a clump or a hole is

{ :

/'*".

/- -"

i@(@i@{wi@\-{.!:. \glgJ-'{gj-l,o'$1!r\i2_Jj "added to the initially symmetric background. The upper,i:i:wli;U

424
-ren

example [Fig. 5(a)] shows a retrograde clump climbing

the vorticity gradient; whereas the lower sequence ofimages tFig. 5(b)l shows a prograde hole moving down
the gradient. Here, time is measured in units of the
background's rotation period, r: l/Cls(r,). Apparently,
there are quite different time scales for these two series.
The clump moves an order of magnitude faster than the{ 1105s{)Fig.5 The vorticity dynamics in two different experimental series: (a) motion of a retrograde clump,starting at the periphery, and (b) motion of aprograde hole, starting at the center. The

repetitive vorticity grayscale is linear. The outerarcs indicate the wall radius B*"u = 3.5 cm. Time ishole.

measured in units of the vortex's rotation period,

4. Analysis
Our experiments with retrograde clump placed
initially at plasma column periphery (ru = Rp) haveconsideration. Figure 6 shows a sequence of measured
radial velocity points for retrograde clumps of different

shown that the clump rapidly (t < i accelerates to an
approximately constant radial velocity, and then spirals
toward the center of background vorticity. The radial

strength {, /r,. ln the plot, the radial velocity isnormalized to r,2 ([ to emphasize the vortex strengthcharacteristics of the motion. The plot also shows thevelocity of the clump increases with its circulation l"
and background vorticity gradient (,i, while it decreases
as the local rotational shear Q[ intensifies. While theresults of linear computations tEq. (a)l and VIC
simulations [9]. There is a good agreement between
experimental results, theory and VIC simulations in theirclump approaches close to the center of the background(r, - p,), within a time of a few orbiting periods it
forms a dipolar structure, which is out of presentintersection region, where 4 lr, << l. In this linear
regime, the radial velocity scales as dr,ldt * (,2((.
r21

Kabantsev A. et al., Experiments and Theory on 2D Electron Vortex Dynamics in Sheared FlowsMoreover, the measured velocity shows the same trend
in the transition region ( lr, - 7.
Experiments with prograde holes clearly showmuch smaller radial drift velocities. A centered hole is

8.1

a.al

an (unstable) equilibrium, but it eventually moves off
axis; in Fig. 5(b), this takes about 602. For comparison
to theory, we consider the hole motion only when it islr, / (i' r,

I

o,Nl

off-axis. When the hole has moved off center by adistance comparable to its size, it moves with a nearly
constant radial velocity, and then spirals down the
background vorticity gradient. As was the case for a

0.awl

0.'(

0.a1

t

$ir,

retrograde clump, the hole's radial velocity scales as
druldt n 12 ((, bu;t it is approximately 20 times slower.
Unlike the clump motion, the hole needs tens of orbital
periods to move radial distance of Rrl2 and reach theFig. 6 Radial velocity drldt for retrograde clumps andprograde holes versus vortex strength I'lr, in
linear theory (solid curve, Eq.4, c = 0.43)' "mixand-move" estimate (dashed line. Eq. 8), linear
simulation (x's), VIC simulations (o's and o's)'

plasma column periphery.

Figure 6 shows the measured radial velocities for
prograde holes of different strengths. The plot also
shows the "mix-and-move" estimate tEq. (8)l and dataand experiments (r's and O's). In the plot, dr"/dt isnormalized to r,2([ to emphasize this speed as afunction of the vortex strength l/r".

from VIC simulations, as well as the earlymeasurements [2] of two small holes moving at theplasma column periphery. Again, like the case ofretrograde clumps, one can see a quantitative agreement
between the experiment, the "mix-and-move" estimate,
and the VIC simulations for the prograde holes. We noteIagain that the hole moves at much slower rate notbecause the hole has a negative vorticity, but because
the hole is prograde with respect to the background6n {xldcml{*ln'u 4

shear flow in our experiments.

,. r-a' elump's position r"(f) w:.i.'t-r ,'

i {
I I
!.,,

I

i r lo'.ro'--o60

r fr6j
4A

tr

28

The experiments also show that a clump can get"stuck" in a region of a small vorticity gradient (Fig. 7).
Here, the background vorticity has a shoulder with low
local gradient ([ and high rotational shear Q[, so thatthe mixing layer gets flattened before the vortex canmove. Thus, the local mixing will not affect anymore
the background vorticity distribution. Eq. (6) predicts
that a vortex will not pass over a flat shoulder in the

;t0
2.5

1.5

r{om}
Fig. 7 Stagnation of the climbing clump motion on a flatshoulder of the background vorticity. Here, thegradient/shear strength parameter $l0it isaveraged over the clump scale. Note that only the

background vorticity distribution if Lrl(, = (6/A6< l.
Figure 7 shows the background vorticity versus radius,
and the radial position of the clump as a function oftime (dots). The clump starts climbing the gradient andthen stops moving radially at r = 1.5 cm, where theperiphery (1 cm < r < 2.5 cm) of the vorticity
distribution is shown here.

the "wake" behind a moving clump can generatesecondary holes and turbulence. Figure 8 shows theto the background vorticity at different times during an
ascending clump motion. The symmetric (0-averaged)
component of the background vorticity has been
subtracted from the raw images, leaving only theasymmetric component. To conserve the total canonicalangular momentum, the moving clump redistributes the
background vorticity, which forms a spiral wake withlow density at its inner side and increased density at its

azimuthally asymmetric components of the perturbation

outer side. The wake gradually spirals outward [Fig.

clump reaches the shoulder with clump-weighted (,{/Ai= 1, and then, in accordance to expectations, it settles on
this shoulder to be eventually dispersed into thebackground.

Even more strikingly, the experiments show thatr22

Kabantsev A. et al., Expenments and Theory on 2D Electron Vortex Dynamics in Sheared Flows-lxtoss'rX

6q ,
,

*..

kl1t'

+
5.8 r
(a) a spiral coilor retrograde with respect to the local background shear:a prograde vortex moves at a much slower rate than alxl0rs't

retrograde vortex. Sufficiently strong shear breaks down*

the "mix-and-move" model and can completely suppress

this vortex motion on a gentle slope of a background*':nvorticity.
A rapidly moving retrograde vortex leaves a spiraldensity wake, and instability of this wake eventually
generates many longJived holes, which contribute to the"turbulent" noise and slow down the relaxation rate.
Finally, we note that there is obvious discrepancy
between the experimental dynamics and the point-likevortex theory computations when the vortex of realfinite size gets close to the background center, where it
forms a dipolar structure and exhibits a slowly decaying
oscillatory motion. This finite scale behavior merits
further study.

!;15.,1 rO) a "'sed'eif holes

Fig. 8 Evolution of the background vorticity during aclimbing clump motion. The images show only
non-axisymmetric perturbations 6((r,0,t) of the
O-averaged background vorticity (o(,if ).

Acknowledgment
This work was supported by National Science8(a)1, and eventually evolves through high m-modes of
the Kelvin-Helmholtz instability into a set of small holes

located near the plasma periphery tFig. 8(b)1. Typicallythese holes are evenly spaced in 0 and have close valuesof ru. The slow drift of these holes out of the vorticity

Foundation Grant PHY-987699.

distribution controls the later stage of relaxation of thisUl D.L. Eggleston, Phys. Plasmas 1, 3850 (1994).
t2l X.-P. Huang, K.S. Fine and C.F. Driscoll, Phys.
Rev. Lett. 74,4424 (1995).
t3l K.S. Fine et a/., Phys. Rev. Leu. 75,3277 (1995).
t4l Y. Kiwamoto et al., Phys. Rev. Lett. 85, 3173

References"turbulence" toward an ordered state. Descending spiralmotion of prograde hole does not form such kind of
wake due to fast and more thorough mixing of the
background vorticity.

(2000).

5. Conclusions

t5l C.G. Rossby, J. Mar. Res. 7, 175 (1948).
t6l R.K. Smith and W. Ulrich, Q.J.R.Meteorol. Soc.
rr9,2o7 (1993).

We have measured the radial drift velocities offinite-sized clumps and holes moving up and down
vorticity gradient for a wide range of vortex strengths,[7] G.M. Reznik and W.K. Dewar, J. Fliud. Mech.269,

background gradients, and background shear rates. All
measurements are in quantitative agreement with recent[8] W.H. Shubert et al., J. Atmos. Sci. 56. 1197

301 (1994).

theoretical analyses for weak vortices, and with(reeg).

numerical simulations.
Clumps and holes move in opposite directions and[9] D.A. Schecter and D.H.E. Dubin, Phvs. Rev. Lett.

at substantially different rates. Both types of vortices
mix the background vorticity and act to level its

ll0l T.H. Dupree, Phys. Fluids 26,2460 (1983).

83,2r9r (1999).

gradient. However, conservation of angular momentum[11] R.J. Briggs, J.D. Daugherty and R.H. Levy, Phys.
Fluids 13, 421(1970); R.H. Levy, Phys. Fluids 11,

dictates that clumps move up the background gradient,whereas holes move down the gradient. The rate of thismotion is determined by whether the vortex is prograde

920 (l968).
[12] C.F. Driscoll and K.S. Fine, Phys. Fluids B 2,1359(1990).

123