Proceedings of ITC/ISHW2007

Cascade process of two-dimensional turbulence observed inmagnetized pure electron plasmas
Yosuke KAWAI1) , Yasuhito KIWAMOTO1) , Yukihiro SOGA1)∗and Jun AOKI1)∗
1)

Graduate School of Human and Environmental Studies, Kyoto University, Yoshida Nihonmatsu-cho, Sakyo-ku, Kyoto
606-8501, Japan(15 October 2007)
Elementary dynamics of two-dimensional (2D) turbulence are examined experimentally by extensive analyses of ﬁne-scale structures in the density distribution of a magnetized pure electron plasma. We observe therelaxation process of the electron plasma involving stochastic mergers of vortex patches generated by the instability in terms of a spatial pattern of vorticity, and compare with the theoretical picture of the 2D turbulence
described in the wave-number (k) space. In the merging process among vortex patches, the energy spectrum
shows a algebraic dependence of k−α above the kin j corresponding to the size of the ﬁrst generated patches. Theenergy and enstrophy tranfer rate show the characteristic features of the cascade process predicted by the theory,i.e., in the ﬁne-length scale of k > kin j the enstrophy cascades upward at a constant transfer rate, and in the regionk < kin j the energy transfers down to lower k. Through time-averaging the evolution of the spectra, we derive
the spectrum in the stationary turbulence sustained by the successive input of vortices due to the instability. The
resultant spectrum is qualitatively consistent with the 2D turbulence theory, but also shows the discrepancies thatthe power index of the energy spectrum is larger than the theoretical prediction of α = 3 and that the enstrophytransfer rate is almost zero around k ≈ kin j reﬂecting the eﬀect of coherent vortices.
Keywords: non-neutral plasmas, two-dimensional turbulence, vortex dynamics, spectral analysis, enstrophy cascade
DOI:

Macroscopic dynamics of pure electron plasmas transverse to a strong magnetic ﬁeld are equivalent to the twodimensional (2D) Euler ﬂuid within the guiding-center approximation, and the electron density n(x, y) and the potential distribution φ(x, y) are proportional to the vorticity
ζ(x, y) and the stream function ψ(x, y), respectively [1, 2].
Taking advantage of this equivalence, magnetized pure
electron plasmas have been employed extensively for detailed examinations of many aspects of 2D vortical dynamics, that constitute the elementary processes of 2D turbulence, such as the advection, merger, ﬁlamentation of vortices [2–4]
Vortex patches are spontaneously generated in thenonlinear stage of the diocotron instability from the ringshaped electron density with a strong radial shear of theazimuthal ﬂow [1, 5]. Free relaxations of the unstable system include stochastic dynamics of vortex patches. Timeresolved spectral analyses have been carried out along therelaxation of the turbulent states, focusing on the particle
transport [6]. In this paper, we extend these examinations
further to explore fundamental properties of 2D turbulenceof the vorticity distribution in terms of the transport of theenergy and enstrophy in the wave-number (k) space.
∗ Present address:

Department of Physics, Graduate School of Science, Osaka University, 1-1 Machikaneyama, Toyonaka, Osaka 5600043, Japanauthor’s e-mail: kwaiyosuke@h01a0395.mbox.media.kyoto-u.ac.jp

Theoretical picture of 2D turbulence have been conjectured by Kraichnan [7] and Batchelor [8]. Both of themproposed that in the isotropic and homogeneous 2D turbulence,the enstrophy injected at the length-scale of lin j/(∝ 1 kin j ) cascades at a constant transfer-rate of η down toa scale of dissipation ld (∝ 1/kd ) and dissipates by viscosity. This cascade picture of 2D turbulence leads to an energy spectrum characterized by the power-law E(k) ∝ k−3
in the inertial range of kin j ≤ k ≤ kd in the wave-numberspace.
The cascade process of the enstrophy has been investigated experimentally by using thin layers of electrolytes [9, 10] and soap ﬁlms [11], and k−3 scaling has
been observed. However, in these experiments, it is diﬃcult to resolve the ﬁne vortical structures extending to thedissipative range, and therefore there remains some uncertainties in comparing the vortex dynamics observed in thereal space with the theoretical picture of 2D turbulence described in the spectral space.
In the experimental investigation with pure electron
plasmas, the vorticity distribution can be observed directlyin terms of the electron density distribution n(x, y) down tothe dissipative scale at a high signal-to-noise ratio. With
this advantage, in this paper, we examine the cascade process of the free-decaying 2D turbulence extensively inwide length scales. Moreover, by considering this phenomena as an elementary process of forced 2D turbulence, weProceedings of ITC/ISHW2007

Density (1013 m-3)

5 Μs

1315

21

23

1

6

1.4

0.9

1.1
0.8
0.5 25!
(10-3 m2 s-1)

531

40

60

200

0.3

30.8

20.7

1

0

00.6Fig. 1 Images of the time evolution of the density distribution.
The darkness is proportional to the density. The time ofobservation (in µs) is indicated at the upper left corner.

derive the spectrum in the stationary state through timeaveraging the evolution of spectra and compare with theKraichnan-Batchelor theory.
The experiment is carried out by containing a pure
electron plasma in a Malmberg-trap with a uniform magnetic ﬁeld (B0 = 0.048 T) and a square-well potential. After the plasma is formed into a ring-shaped proﬁle, the density distribution at a speciﬁc time in the relaxation process
is observed by damping the electrons onto the conducting
phosphor screen and recording the luminosity distribution
with a 512 × 512 pixel CCD camera providing the spatial resolution of 0.1 mm/pix. This destructive diagnostic
requires a high reproducibility of the initial proﬁles, because the relaxation process of the turbulence triggered by
the instability is stochastic in nature. Therefore in this experiment, in addition to technically minimizing shot-byshot variations in the initial proﬁles, an ensemble-average
is applied over typically 5 shots of data for each time ofthe observation in examining the time evolution of physical quantities characterizing the turbulence. The details of
the experimental conﬁguration and method are reported inRefs. [4, 6, 12, 13].
The time evolution of the observed density distribution is shown in Fig. 1. The ring proﬁle is produced at 5µs, and distorted by the diocotron instability within a fewµs. The deformed distribution grows on the fastest growing diocotron mode and eventually 5 high-density vortex
patches are generated at 13 µs. After the formation of theﬁrst vortex patches, the number of patches decreases successively through intermittent mergers among the patches.
This stochastic process is accompanied by the generationof ﬁlamentary structures in the low density part of the population that extend the length-scales to ﬁner regions. The
outward transport of the ﬁlamentary structures is accompanied by the inward transport of the high-density patches as
observed at 31 µs. The concentrated patches rotating withthe period of 10 µs expel ﬁlaments from the central region,and ﬁnally form a bell-shaped core distribution that is surrounded by a low density halo with striations (t = 200 µs).
Turbulent states are characterized by areal integrals of
the vorticity (∝ n(x, y)). Figure 2 plots some of the integrals calculated from the measured density distributionn(r, θ) as a function of time. Each integral is normalized to!th(2!10-3 m2 s-1)

451020 3050100

5102050

100 200200time ("s)

Fig. 2 The time evolution of energy E (◦), total electron numberN (♦), angular momentum L (△), enstrophy Z2 () andpalinstrophy P (H). Inset: The time evolution of the viscosity coeﬃcients estimated experimentally (◦) and calculated from Eq. (62) in Ref. [16] (•).

unity. The integrals include∫ the electrostatic energy (ﬂuidkinetic energy) E = 1/2 d2 rn (−eφ),the total electron∫
2
number (total circulation) N = ∫ d rn, the angular momentum ∫(angular impulse) L = d2 rnr2 , the∫ enstrophy
Z2 = 1/2 d2 rn2 and the palinstrophy P = 1/2 d2 r |∇n|2 .
The palinstrophy is a measure of the ﬁne-scale structuresin the turbulence [8, 9, 14].
Throughout the whole process, E, N and L do notshow any systematic change except 5 % variations probably attributable to the shot-by-shot ﬂuctuations in the generation process of the unstable initial distributions. Therefore the three integrals may be considered to be invariant. In contrast, Z2 and P show unambiguous systematicchanges. The enstrophy Z2 undergoes a substantial decaying through the merging processes, and ﬁnally goes down
to 40 % of the initial value. The palinstrophy P shows arapid increase while mergers among the vortex patches areactive. It is maximized at 31 µs when the ﬁlamentary structures are conspicuous in the region outside the high-density
core as shown in Fig. 1. After the maximization, P drops
steeply concomitantly with the reduction of the decreasing rate of the enstrophy, suggesting the manifestation ofdissipative eﬀects in ﬁner length-scales.
In order to estimate the degree of enstrophy dissipation in this experiment, we introduce a Newtonian viscosity into the 2D Euler equation as a correction, i.e.,
Dn/Dt = ν∇2 n, where ν is the kinematic viscosity. (Notethat n is propotional to the vorticity.) This equation leads
to the relation of the decreasing rate of the enstrophy to theviscosity and palinstrophy as follows [8, 9, 14]:
DZ2= −2νPDt(1)

By introducing the experimental values of Z2 and P into
Eq. (1), the eﬀective viscosity ν is evaluated and plotted asa function of time in Fig. 2. ν is maximized at time t ≈ 23
µs when the density conﬁguration changes drastically fromseparated vortex patches to a single-peaked distribution.
The collisional transport of a magnetized pure electron plasma in the 2D regime [15] has been studied an-Proceedings of ITC/ISHW2007(E(k) =1e2 ε0 B0)2 ∫ 2π
kdϕ0|n(k)|2,k2(2)where ϕ is the azimuthal angle of k. The time evolutionof E(k) thus obtained is shown in Fig. 3. In the initial
distribution at 5 µs, the energy spectrum shows oscillatory structures due to the concentration of the energy at
the ring radius. When the ring distribution is torn into thevortex patches at 13 µs, the spectrum has a local maximum
around the wave number k = kin j ≈ 500 consistent with thesize of the ﬁrst vortex patches. Along with the subsequent
mergers between patches (t = 13 ∼ 31 µs), the location of
the spectral maximum shifts progressively down to lower
wave numbers, and the dip around k ≡ kcore ≈ 300 whichcorresponds to the core size in the ﬁnal state is ﬁlled.
In length-scales smaller than the width of the vortex
patches k > kin j , the energy spectrum broadens upward
and the E(k) exhibits a power-law dependence of ∝ k−α .
The slope of the spectrum in the interval 700 ≤ k ≤ 5000
is drawn in Fig. 3. Throughout the merging processes, thepower index α remains around 5, apparently larger than
the theoretically predicted value of 3 [8]. The upper limit
in the wave-number space k = kd to which E(k) shows apower-law scaling reaches its maximum value of ≈ 8300
at 31 µs when the palinstrophy is maximized. This scalelength (≈ 0.38 mm) is consistent with the thickness of theﬁlamentary structure at the end of spiral arms displayed inFig. 1.
In the slow relaxation process toward the asymptotic
state after the merger (t = 40 ∼ 200 µs), the energy concentrates at kcore and decreases steeply beyond k ≈ 1000. Thepower index in the high wave-number region 1000 ≤ k ≤5000 decreases slowly from ≈ 5 toward 3.5. The powerlaw spectrum at high k represents ﬁne structures remaining
in the halo region surrounding the high density core.
The rate of upward energy transfer ε(k) through k isevaluated from the time-resolved energy spectrum in Fig. 3as∫ k∂E(k).
(3)
ε(k) = −dk∂t0

5 Μs102

E(k) (m2 s-2)

alytically, in which the viscosity coeﬃcient is predicted
as Eq. (62) in Ref. [16]. By introducing the parametersof the present experiment to the proposed formula, weestimate the viscosity coeﬃcient νth and plot the resultsin Fig. 2. Though the present study is not under quasistationary states as assumed in the theory, the experimental
evaluation agrees within a factor of 2 with νth . This agreement may suggest that the stochastic motions of individual
particles under ﬂuctuating ﬁelds play a important role inthe dissipation process of vortex dynamics in ﬁne scales.
To compare the experimental results to the theoretical
picture of 2D turbulence, we calculate the energy spectrum
in k space from the measured density distribution. The energy spectrum E(k) is determined from∫ the Fourier transform of the density distribution n(k) = d2 re−ik·r n(r) as

13

15

21

23!5.

!5.2!4.8!5.1

31!3.840

60

200!3.8!3.8!3.5

10-2

25

102

!4.8

10-2
102

103

102

103

102

103

k (m-1)

102

103

102

103

Fig. 3 The time evolution of the energy spectrum calculated
from the measured n(r). Numbers at the upper-left cornerstand for the time of the observation.
"(k) (2!108 m2 s-3)
232 17Μs!(k) (1014 s-3)
2527

31

0-2
102

103

102

103

102

103

k (m-1)

102

103

102

103

Fig. 4 The time evolution of the upward transfer rates of theenergy ε(k) (dashed line) and enstrophy η(k) (solid line)
through k.

The enstrophy transfer rate η(k) is evaluated similarly fromthe enstrophy spectrum Z(k) = k2 E(k). The time evolutionof ε(k) and η(k) during the period in which E(k) follows
the power-law is shown in Fig. 4. The observation of ε(k)indicates that the energy is transfered downward and therate of transfers is maximum around kcore at each time. Incontrast, the enstrophy is transferred upward in the wavenumber space with k ≥ kin j . In particular, over the wide
range of k ≥ 1500, η(k) is almost constant as assumed in
the theoretical picture of 2D turbulence. Both in the energy
and in the enstrophy, the transfer rates are maximized at t ≈25 µs when the density conﬁguration changes drastically.
The observation that η(k) decreases to zero at k < 2kin jfrom a constant value at k > 3kin j corresponds to the observed vortex dynamics in Fig. 1, i.e., the vortex patch retain its shape for a long time with high vorticities (enstrophy) and the ﬁlamentation of the structure dominates in theoutside region of the patches.
The time-resolved data presented above include details of free-decaying 2D turbulence whose enstrophy isfed at k = kin j corresponding to the ﬁrst generation of vortex patches and that is left in an isolated state. Here wetry to construct the energy spectrum in a stationary statethat is maintained by continuous energy input at k = kin j
due to the instability and by the continuous dissipation atk ≫ kin j by using the above data as an elementary process constituting a stationary turbulence. If the interaction
among the structures appearing in diﬀerent stages of thefree-decaying turbulence is negligible, the time-averaged
spectra, Ē(k), ε̄(k) and η̄(k), may approximately represent
the characteristic features of the stationary turbulence. Theresults of the time-weighted average of the observed data
are summarized in Fig. 5.
The constructed spectra exhibit characteristics close
to the fundamental features of stationary 2D turbulence:Proceedings of ITC/ISHW2007
2!(k) (4!1012 s-3)0E(k) (m2 s-2)102"(k) (107 m2 s -3)

-2

100

102

103

k (m-1)

4.4

10-2

104

10-4
102

103

k (m-1)

104

Fig. 5 The energy spectrum Ē(k) time-averaged in the interval
t = 13 ∼ 200 µs. Inset shows the time-averaged transfer
rates of the energy ε̄(k) (dash line) and enstrophy η̄(k)(solid line).

The enstrophy is transfered upward at a constant rate abovek ≈ 3kin j . With this transfer rate, the dissipative wave
number is estimated at kd ≈ 7200 from the expression
ld ≈ η−1/6 ν1/2 proposed in Ref. [8]. Figure 5 also showsthat Ē(k) depends algebraically on k in a wide range of thewave-number space kin j ≤ k ≤ kd . In contrast to η̄(k), it
is conﬁrmed that in the region k < 3kin j , the energy ﬂux
ε̄(k) proceeds toward small wave-numbers and is maximized around kcore corresponding to the size of the core
distribution in the asymptotic state.
Through the region k ≥ 3kin j where the enstrophytransfer rate is constant, the energy transfer rate is observed
to be zero. This observation supports the theoretical expectations that the spectral dynamics at high wave-numbers ofthe 2D turbulence are governed by the enstrophy cascade
process [7]. On the other hand, in the region k < kin j , theobservation that η̄(k) ∼ 0 and ε̄(k) < 0 indicates that thedynamics in the energy spectra obey the inverse cascade
process. The observed non-uniformity of ε̄(k) is understood in terms of the absence of a dissipation mechanism
at large length-scales in the strongly magnetized pure electron plasma.
In the intermediate region kin j ≤ k < 3kin j , neitherε̄(k) nor η̄(k) is zero, indicating that the transfer exists inthe k space of both the energy and enstrophy. The local
non-uniformity of ε̄(k) and η̄(k) in this region suggests thebreak-down of the ubiqitous cascade model. This observation is closely related to the persisting presence of coherent vortices that capture a large amount of enstrophy
and inhibit the cascade as observed by numerical simulations [17, 18]. This is probably the reason why the powerindex of the observed energy spectrum Ē(k) in the inertial
range of kin j ≤ k ≤ kd is α = 4.4 and larger than the theoretical prediction of α = 3 [7].
In this paper, we have examined the vortex dynamics of 2D turbulence in a magnetized pure electron plasma
over a wide range of length-scales extending from the injection scale down to the dissipative scale, and compared

the experimental results to existing theories of 2D turbulence. In the stage characterized by the successive mergers between vortex patches starting from the unstable initial density proﬁle, the observed density distribution exhibits turbulent characteristics. While the energy is transfered downward, the enstrophy undergoes an upward transport starting from the injection wave number kin j . In ﬁner
length-scales with k ≥ 3kin j , the transfer rate of the enstrophy is observed to be constant, and the energy spectrum
shows a power-law scaling E(k) ∝ k−α in a broad inertial
range with α ranging from 5.2 to 3.5. The discrepancy
from the theoretically-expected value of α = 3 is attributed
to the inhibition of the cascade process reﬂecting the eﬀectof the long persistence of high-vorticity patches.
The author thanks Professor K. Itoh of NIFS for enlightening discussions. This research was supported by
the Grant-in-Aid for Scientiﬁc Research (B) 17340173 ofJSPS and partly by the collaborative program of NIFS.
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