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22 IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 30, NO. 1, FEBRUARY 2002
Generation and Instability of Spiral Wakes in
Sheared Electron Flows
Andrey A. Kabantsev and C. Fred Driscoll
Abstract—Images of the electron density of a magnetized elec-
tron columnillustrate the dynamics of intense vortices moving in a
weaker background vorticity. A moving retrograde clump of vor-
ticitygeneratesaspiralwake thatthenevolvesintomanylong-lived
holes, contributing to the late time fluctuations.
Index Terms—Electron plasmas, rotational shear, two-dimen-
sional vortex dynamics, turbulent rotational flows, vortex.
M
ANY FEATURES of turbulent relaxation in two-di-
mensional (2-D), nearly inviscid, incompressible
flows have been studied using electron plasmas confined in
Penning–Malmberg traps. With low viscosity, circular free-slip
boundary conditions, simple manipulating technique, and
accurate diagnostics, magnetized electron columns provide
quantitative tests of theory. The interaction of intense vortices
with a background vorticity gradient plays an important role
in 2-D hydrodynamics, including various aspects of relaxation
toward an ordered state and self-organization of turbulence.
By employing CCD-imaging diagnostics, we examine key
elementary processes in such systems.
Detailed description of the experimental apparatus may be
found in [1], [2]. In the experiment, we first inject and trap
a stable symmetric electron column with initial density pro-
file
. The trapped column typically has maximum den-
sity
10 cm , characteristic radius 2 cm, axial
length
35 cm, and electron thermal energy 1eV.
The column
drift rotates with angular velocity ;
typically,
decreases with , so the column has negative shear.
This causes the clump to be classified as a retrograde vortex.
The axial bounce time of individual electrons (
1 s) is short
compared with the bulk column rotation time (
50 s), so the
electrons effectively average over any
variations.
The
drift flow in plane of these electrons is
mathematically isomorphic to the
flow of vorticity in an
incompressible inviscid fluid [3]. Here, the electron density cor-
responds to vorticity, and the electrostatic potential corresponds
to the stream function with free-slip boundary conditions.
We study the motion of intense vortices on a weaker back-
ground vorticity by combining two separate electron columns
of different densities and sizes. We can easily create clumps
characterized by approximately “Gaussian” density (vorticity)
distributions, with maximum density comparable to the back-
Manuscript received July 2, 2001. This work was supported by the National
Science Foundation under Grant PHY-9876999 and by the Office of Naval
Research under Grant N00014-96-1-0239.
The authors are with the Department of Physics, University of California,
San Diego, La Jolla, CA 92093 USA (e-mail: fdriscol@uscd.edu).
Publisher Item Identifier S 0093-3813(02)03160-0.
ground column density, and with radial extent . The
column with a clump then evolves during a time
, after which
it is dumped axially onto a phosphor screen biased to
15 kV.
The 2-D density image
is recorded with a low-noise
CCD camera with pixel area of (.13 mm)
on the screen. Al-
though this imaging technique is destructive, the shot-to-shot
variations for nearly identical initial conditions are small (
1%
in azimuthally averaged local density), so the time evolution of
a flow can be studied.
Our experiments with retrograde clumps placed initially at
plasma column periphery
have shown that the clump
rapidly accelerates to an approximately constant radial velocity,
and then spirals toward the center of background density dis-
tribution. The measured rates of this motion are in quantitative
agreement with theoretical predictions and numerical simula-
tion [4]. After the clump reaches the center of the background
, it forms a stable dipolar structure.
The experiments also show that the spiral wake behind a
moving clump can generate “a sea” of secondary self-trapped
holes, giving turbulence; this is not understood theoretically.
Fig. 1 shows the initial
of the background column
combined with the intense clump; and images of the az-
imuthally asymmetric components
of the density
perturbations at three successive times during an ascending
clump motion. Here, the
-symmetric component of
the background electron density has been subtracted from the
raw CCD-images, leaving only the asymmetric perturbation.
The moving clump redistributes the background density as it
moves inward, which forms a spiral wake with low density at
its inner side and increased density at its outer side [4]. This
wake rotates differentially with respect to the clump, as seen
in Fig. 1. The space charge density induced by the spiral wake
on the background electron column generates an azimuthal
electric field at the position of the clump and drives it radially.
This wake shows long-lasting behavior, and it remains long
after the clump has reached the center of the background. The
wake gradually spirals outward, and also evolves through high
-modes of the Kelvin–Helmholtz instability into an ordered
set of small holes located near the plasma periphery. Typically
these holes are evenly spaced in
at close radii. The slow drift
of these long-lived holes out of the vorticity distribution con-
trols the later stages of relaxation of this secondary small-scale
turbulence. The slow outward spiral of the prograde holes does
not form a strong wake due to the more thorough mixing of the
background density distribution by the prograde vortex.
These images show a new nonviscous mechanism of effec-
tive energy transfer from the large-scale to the small-scale tur-
0093-3813/02$17.00 © 2002 IEEE
KABANTSEV AND DRISCOLL: GENERATION AND INSTABILITY OF SPIRAL WAKES IN SHEARED ELECTRON FLOWS 23
Fig. 1. Initial density distribution and density perturbations left by an ascending clump at three successive times. The positive/negative color scales are shown at
the right of each figure(densities have units of 10
cm ). The background column rotates counterclockwise. Time is measured in units of the background rotation
period,
2 . The outer circles show cylindrical wall of the trap, 3.5 cm.
bulence and provide additional insight into the nature and de-
velopment of relaxation processes in turbulent plasma flows.
R
EFERENCES
[1] X.-P. Huang, K. S. Fine, and C. F. Driscoll, “Coherent vorticity holes
from 2-D turbulence decaying in a background shear flow,” Phys. Rev.
Lett., vol. 74, pp. 4424–4427, 1995.
[2] K. S. Fine, A. C. Cass, W. G. Flynn, and C. F. Driscoll, “Relaxation
of 2-D turbulence to vortex crystals,” Phys. Rev. Lett., vol. 75, pp.
3277–3280, 1995.
[3] C. F. Driscoll and K. S. Fine, “Experiments on vortex dynamics in pure
electron plasmas,” Phys. Fluids B, vol. 2, no. 6, pp. 1359–1366, 1990.
[4] D. A. Schecter and D. H. E. Dubin, “Vortex motion driven by a back-
ground vorticity gradient,” Phys. Rev. Lett., vol. 83, pp. 2191–2194,
1999.
