J. Plasma Fusion Res. SERIES, Vol.2 (1999) 4849

Self-Organization and Beltrami Conditions
YOSHIDA Zensho' and SUZUKI Takahiro'
Graduate School of Frontier Sciences,lGraduate School of Engineering, University of Totcyo, Totcyo 113-8656, Japan(Received: 9 December 1998 / Accepted: 5 August 1999)

Abstract
A plasma flow induces mixing of magnetic flux, and the length-scale cascades toward a small scale,
resulting in amplification of the magnetic field. If the enhanced Lorentz force becomes to dominate thedynamics, its back-reaction should be taken into account. Once a Beltrami field is generated, it can staystable and stationary against the stretching effect of the plasma flow. The Beltrami condition, which reads
as the force-free condition, imposes a lower bound for the length-scale, avoiding the scale reduction

down to the resistive regime. A statistical model of current filaments accounts well for the spectral
structure of intermittent fluctuations of local currents in experiments.

Keywords:
Beltrami field, intermittency, filamentary structures

1. Introduction

accounts well for the spectral structure of theThe Beltrami condition describes the alignment of
the vector field and its own vorticity. In many different
convective nonlinear systems, the Beltrami condition
plays an important role in characteriztng self-organizedfluctuations. We consider a Boltzmann distribution of
the size of the filaments that maximizes the entropy foran ensemble defined by the total current. An interesting
assertion is that the time series produced by a randommotion model of such filaments generates a power-law
spectra which agrees well with the observation [4].

structures [1,2]. A plasma flow induces mixing ofmagnetic flux, and the length-scale cascades toward asmall scale, resulting in amplification of the magnetic

field. If the enhanced Lorentz force becomes to

2. Beltrami Condition

dominate the dynamics, its back-reaction must be taken
into account. The Beltrami condition, which reads as themagnetic force-free condition, must apply to slow
motion of a strongly magnetized plasma, i.e., themagnetic field B must satisfy V x B = ,1,-B. When ,1, is aconstant over a localized current, this relation imposes alower bound for the length-scale. This bound avoids

Due to the complex mixing process, the current
tends to concentrate in small volumes, which may bedisconnected. When the sectional length-scale of such a
volume becomes small enough, the Lorentz force
dominates. We consider such a "clump" of the magnetic
field that is created by the kinematic dynamo process.
For an arbitrary shape of the clump, the Beltrami
condition V x B = ),8 can be satisfied with a constant ,1,such that l-r: < A < p1, where p* are the eigenvalues ofthe self-adjoint part of the curl operator in the domain Oof the clump [5]. This O is assumed to be multiply

scale reduction down to the resistive regime, andextends the life-time of the amplified magnetic field [3].

In laboratory experiments, we observe that local currents
in a turbulent plasma fluctuate intermittently, suggestingthat the current density has a strongly inhomogeneousdistribution. A statistical model of current filaments

connected (genus ge4 1) and bounded. The eigenvalues@1999 by The Japan Society of Plasma
Science and Nuclear Fusion ResearchC orre sponding author's e -mail: yoshida@ k.u-tokyo.ac.j p48

Yoshida Z. et al., Self-Organization and Beltrami Conditions

are shown to be countable, and are numbered as ...< /r_z3F:<0<p' < 1t23.... Themagnitudesof p*areof-2l0

the order of the cross-sectional length scale of O. The
force-free magnetic clump in the range of 1t_, < ), < 1t,
is stable against internal kink modes, while the stabilityof external modes (deformation of O) depends onA€3-* l0

exterior conditions. Once it is created, the internal
magnetic field is maximally smooth, and no smaller
internal structure can develop in the clump.

l0I

10100

frequency [kHz]3. Statistical Model of Filaments
While each filamentary flux tube, such as aFig. 1 Power spectrum ofi(f) produced by the statisticalmodel, in comparison with the experimentalspectrum. We apply different normalizations toboth spectra to plot them at different vertical

Beltrami field, can be locally stable, their movement andmutual interactions are very complicated, and hence,
they invoke a statistical mechanical treatment. Let us
consider a system of current filaments. Each filament isdenoted by an index m (m - l, ..., N; N is the total
number of filaments). We specify the current density ./on each filament. We can invoke the analogy of thestandard statistical mechanics of particles, wherc J^
parallels the energy level of the eigenstate m. The crosssection o- of the filament la is the statistical variable ofpositions.

o.). The probability of measuring a filament m, at a
fixed observation point, is proportionalto o^.
Figure I shows the frequency spectrum ofj(r)generated by the theory, in comparison with theexperimental spectrum. We observe that the theoretical
spectrum agrees well with the experimental one. Thespectrum ofj(t) has two distinct frequency ranges; thespectrum in the low frequency range (f < f" = 20kHz) is"white", while in the high frequency range (/>f,), thespectrum is the lf spectrum. In the theory, the criticalfrequencyf, is given by f" = l/(lu\t.) = ul(2np"), wherep" is the average radius, A/" is the average duration of afilament, and u is the average speed (random walk) ofthe present model (o. may be regarded as the number ofparticles allocated to the eigenstate m). Each filament
has the current of I^= J.o^.The total current is given
by 1 = DItI^.A micro-state is characterized by

specifying (, = {or, oz, ..', oul. The probability of a

micro-state I is denoted by p( l.).
The expectation value of the current is given by (1)= 2 tp( (, ) I( (, ), where /( I ) is the current of themicro-state I . Maximizing the Shannon entropy ^S =-2 tp( (, ) ln p( I ) for the above-mentioned canonicalfilaments. The two different frequency ranges of thespectrum can be explained as follows. In a long timescale ( | tt - bl > N), j(t) and j(t) come from different
randomly-selected filaments. Therefore, we obtain awhite spectrum. Agreement of our theoretical result withthe experimental spectrum in this frequency range

ensemble, we obtain p( | ) = exp(-9l^J^o^)/2, whereZ = exp(a) =2 texpGFI( { )) z is the partition functionof the canonical distribution.
We assume that the probability for a filament "la"to have a cross-section on (denoted by p^(o^)) is
independent to the all other filaments. Then, p(os oz,justifies our assumption of random movement offilaments.

...,o.v) can be broken down into the simple product ofp^(o) (m = 1,..., N), and we obtain the "Boltzmann

References

Ul Z. Yoshida, Beltrami functions: A paradigm ofdistribution"

streamline chaos, in Current Topics in the Physicsof Fluids,155 (1995).
[2] S.M. Mahajan and Z. Yoshida, Phys. Rev. Lett. 81.,4863 (1998).

p-(o-)=exPGBJ^o^)z^Z^=2 eBJ.o^).

(1)

6_Using the probability distribution (l), we generate[3] Z. Yoshida, Phys. Rev. Lett.77,2722 (1996).
t4l T. Suzuki and, Z. Yoshida, Statistical model ofcurrent filaments in a turbulent plasma (inthe time series of the current density j(r) that ismeasured at a fixed position. Note that the probability ofmeasuring the filament m differs from p.(o.) (theprobability of the filament m to have a cross section

preparation).

[5] Z. Yoshida and Y. Giga, Math. 2.204,235 (1990).
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