数理解析研究所講究録
1121 巻 2000 年 221-228

221

A Statistical Model of Vortex Filaments

Zensho YOSHIDA
University of Tokyo, Graduate School of Frontier Sciences
吉田善章
東京大学大学院新領域創成科学研究科

1

Introduction

The Beltrami condition describes the alignment of the vector field and its own vorticity.
In many different nvective nonlinear systems, such as fluids and plasmas, the Beltrami
condition plays an important role in characterizing self-organized structures.
A plasma flow induces mixing of magnetic flux. The length scale cascades toward a small
scale, resulting in amplification of the magnetic field. If the enhanced Lorentz force becomesto dominate the dynamics, its back-reaction must be taken into account. The Beltrami
condition, which reads as the magnetic force-free condition, must apply to slow motionof a strongly magnetized plasma, i.e., the magnetic field $B$ must satisfy $\nabla\cross B=\lambda B$ .
This relation characterizes the local structure of stable vorteses. In this paper, we derivea Boltzmann distribution of the size of the filaments that maximizes the entropy for an
ensemble defined by the total current. An interesting assertion is that the time series
produced by a random-motion model of such filaments generates a power-law spectra which
agrees well with experimantal observation [1].
$\mathrm{c}o$2Intermittency

Intermittency is one of the common issues in various complex systems. The underlying
mechanism may be either low-dimensional chaotic dynamics or high-dimensional statistical
dynamics. A typical example of the latter category is the fluid turbulence [2]. When aturbulent flow has filamentary structures of vortex tubes, local measurements of the flow,or other related physical quantities, detect intermittent fluctuations. Three-dimensional
shear flows stretch vortex tubes and produce thinner tubes carrying stronger vorticities(Kelvin’s circulation law), and hence, the vorticity ( $\omega=\nabla\cross v;v$ is the flow) tends toconcentrate into filamentary tubes.
In plasmas, the induction effect and the corresponding Lorentz back reaction bring about
coupling between the flow and the magnetic field $B$ , resulting in a considerable complication of the dynamics. The stretching effect applies also to the magnetic flux, and $B$ tends
to take irregular distributions. There are many different examples where various physical
$v$

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quantities exhibit intermittent behavior and filamentary structures. In the solar photosphere, magnetic field strengths vary in the range of to 0.2 T. The characteristic length. Satellite measurements
scale of the magnetic field distribution is of the order ofof the magnetotail of the earth detect intermittent fluctuations in ion bulk velocities andmagnetic fields [4].
We consider the filamentation of the electric current $j=\nabla\cross B/\mu_{0}(\mu_{0}$ ; vacuum permeability), which is the vorticity of the magnetic field in a plasma. The reduction of thelength scale of the magnetic field has been discussed by many authors in its relation to
the turbulent cascade process and self-organizations $[5, 6]$ . The induction effect due toturbulent plasma flow brings about stretching of flux tubes and generates locally-stable
flux filaments [6]. Such filamentary structures of magnetic fields resemble those of vortices
created in three-dimensional turbulent flows [7]. While each filamentary flux tube, such asa Beltrami field [6], can be locally stable, their movement and mutual interactions are very
complicated [8], and hence, they invoke a statistical mechanical treatment.
Some experimental observations also give us a motivation to develop a model of strongly
irregular current in a turbulent plasma. Measurements of local currents in turbulent plasmas show intermittent fluctuations. In a reversed-field-pinch plasma, internal currents
were measured by inserting a small electrostatic energy analyzer [9]. The data exhibit
strong fluctuations, suggesting that the current density has strongly inhomogeneous distributions [10]. We observe similar behavior on REPUTE-I [11] (Fig. 1).
$0$$100\mathrm{k}\mathrm{m}[3]$$\overline{\mathrm{b}\sim\triangleleft}$$\underline{\Xi}$$.\wedge-\vee\neg$

UIIE$\mathrm{L}^{\mathrm{l}11\mathrm{O}}\mathrm{J}$T. Suzuki and Z. Yoshida, Fig. 1

Figure 1: The waveform of the local current density (toroidal component) measured inside
the plasma. The distance of the detector from the center is 75% of the radius of the plasma
column. The detector is directed parallel to the local magnetic field during the flat top
phase of the discharge.

Fluctuations in the local current density are more than 100% of the mean current density,

223
while the system is in a macroscopic quasi-steady state. Such a system is insuperable fromthe viewpoint of conventional theories such as the diffusion model of the current density
or the magnetohydrodynamic relaxation model [12].
3

Statistical Model of Filaments

We apply a statistical mechanical method to characterize the spectral structure of thecurrent-density fluctuations. The principal macroscopic parameter that characterizes thesystem is the total current, which is a controllable stationary parameter. We note that we
do not invoke “energy” to characterize the system. We are not discussing the dynamical
process of generating filaments or the dynamics of many filaments, and hence, the energy
arguments do not apply. Because of the resistive dissipation, the energy is not in equilibrium. The filamentary structure of the current yields an enhanced dissipation. In such a“dissipative structure”, fluctuations creating filaments yield large dissipation, so that the
principle of the minimum energy dissipation is invalidated.
The total current is the integral (sum) of local currents (filament currents). We study
the most probable statistical distribution of filament currents for a given total current.
Theoretical predictions will be compared with experimental observations.

T. Suzuki and Z. Yoshida, Fig. 2

Figure 2: A model of current filaments in a plasma column.

Let us consider a system of current filaments. Each filament is denoted by an index( $m=1,$$N;N$ is the total number of filaments in the ensemble). We specify the current
densityon each filament. The toroidal component is given by $J_{m}=J_{m}\cdot n$ , whereis the unit vector in the toroidal direction. We can invoke the analogy of the standard
statistical mechanics of particles, whereparallels the energy level of the eigenstate .
The cross sectionof the filament is the statistical variable of the present model
may be regarded as the number of particles allocated to the eigenstate ). Each filament
has the current of(see Fig. 2).

$m$$\cdots,$$J_{m}$$n$$J_{m}$$\sigma_{m}$

$m$

$m$

$(\sigma_{m}$$m$$I_{m}=\sqrt\sigma mm$

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We consider a sub-domain of the total cross section of the plasma column as to be acanonical ensemble that contains $N$ filaments. This $N$ may change, as a grand canonicalensemble; however, it is appropriate to assume that the “chemical potential” is zero. The
total current in is given by$\Omega$$\Omega$$($$I= \sum_{m=1}^{N}I_{m}=\sum m=N1J_{m}\sigma_{m}$A micro-state is characterized by specifying
micro-state is denoted byby imposing(1)$\ell\equiv\{\sigma_{1}, \sigma_{2}, \cdots, \sigma_{N}\}$$p(\ell)\equiv p(\sigma_{1}, \sigma_{2}, \cdots , \sigma_{N})$$\ell$.

. The probability of a

. We have to normalize the probability$P= \sum_{\ell}p(\ell)\equiv 1$(2),

where the summation is taken over all possible micro-states. The expectation value of thecurrent is given by(3)$(I)= \sum_{\ell}p(\ell)I(\ell)$ ,where $I(l)$ is the current of the micro-state ;$\ell$.

(4)$I( \ell)\equiv I(\sigma 1, \sigma 2, \cdots, \sigma_{N})=\sum_{m}J_{m}\sigma m$

We derive the probability distribution by maximizing the Shannon entropy$S=- \sum_{l}p(l)\ln p(\ell)$.

(5)

The use of the Shannon entropy is appropriate when we can assume that the whole system of filaments are statistically homogeneous (in a more restricted system, we use the
R\’enyi entropy). Apparently, some filaments are connected through turns around the torus.
However, we neglect the long-range correlations through toroidal turns, assuming that thecurrent channels are strongly chaotic. For the above-mentioned canonical ensemble, we

maximize$H=S-\alpha P-\beta\langle I\rangle$,(6)

where and are Lagrange multipliers. The reciprocal of represents the “temperature”
of fluctuati . The variation principle $\delta H=0$ yields the canonical distribution$\alpha$$\beta$$\beta$$o\mathrm{n}\mathrm{s}$$p(\ell)=\exp(-\alpha-\beta I(\ell))$.

(7)

The normalization condition (2) determines ;$\alpha$$Z \equiv\exp(\alpha)=\sum_{\ell}\exp(-\beta I(\ell))$.

(8)This is the partition function of the canonical distribution.
The probability (7) now reads$Z$$p( \ell)\equiv p(\sigma 1, \sigma_{2}, \cdots,\sigma_{N})=\frac{\exp(-\beta\Sigma_{m}Jm\sigma m)}{Z}$

.

(9)

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We assume that the probability for a filament “ ” to have a cross-section(denoted byis independent to the all other filaments. Then,
can be broken
down into the simple product of $p_{m}(\sigma_{m})(m=1, \cdots, N)$ , and we obtain the “Boltzmann
distribution”
,
(10)
$m$

$\sigma_{m}$$p_{m}(\sigma_{m}))$$p(\sigma_{1}, \sigma_{2}, \cdots, \sigma_{N})$$p_{m}( \sigma_{m})=\frac{\exp(-\beta Jm)\sigma_{m}}{Z_{m}}$where$Z_{m}= \sum_{\sigma_{m}}\exp(-\beta J_{m}\sigma m)$

The inverse temperature
we observe

.

(11)

characterizes the intensity of fluctuations. For a certain ,$\beta$$\sigma$$\frac{p_{m}(\sigma)}{\Sigma_{k}p_{k}(\sigma)}$$=$$\frac{Z_{m}^{-1}\exp(-\beta J_{m}\sigma)}{\Sigma_{k}Z_{k}^{-1}\exp(-\beta J_{k}\sigma)}$$=$$\frac{1}{1+\Sigma_{k}\neq m(Zm/zk)\exp(-\beta\sigma(Jk-J_{m}))}$$\equiv$$\frac{1}{1+A_{m}}$

.

(12)

In the limit of zero temperature$A_{m}=0$ for the filament $m$ that satisfies$J_{m}= \min_{k}(J_{k})$ , and $A_{m}=$ oo for other filaments.
This implies that the whole current
condensates on the “grand state”, that is the filament which has the minimum current
density. Thismust be the average current density over the cross section , andthen, is the total area of . Therefore, the current distributes homogeneously on .
By calculating the total ohmic dissipation produced by a local resistivity , we easilyfind that the effective resistivity for the mean current is given by$(\betaarrow\infty),$$\sqrt\min$$J_{\min}$$\Omega$$\Omega$$\sigma$$\Omega$$\eta$$\eta^{*}=\eta\frac{(J^{2}\rangle}{\{J)^{2}}$In the limit of4$\betaarrow\infty,$$\eta^{*}=\eta$, while a finite$\beta$.

(13)

yields an enhanced effective resistivity.

Time Seriese Generated by Ramdom Motion of Filaments

Using the probability distribution (10), we generate the time series of the current densitythat is measured at a fixed position in . Note that the probability of measuring thefilament $m$ differs from(the probability of the filament $m$ to have a cross section. We denote by the probability of measuring a filament at a fixed point, which is
proportional to. Here, we assume that the filaments cover , and set$j(t)$$\Omega$$p_{m}(\sigma_{m})$$\sigma_{m})$$m$$q_{m}$$\Omega$$\sigma_{m}$$q_{m}= \frac{\sigma_{m}}{|\Omega|}$

,

(14)

where
is the area of .
To evaluate the period of a certain filament being measured at the observation point, we
specify the speed , which is perpendicular to the toroidal direction, of the filament. We
assume that is common for all filaments. Let
be the radius of the filament ;$\Omega$$|\Omega|$

$v$

$v$

$m$

$\rho_{m}$$\rho_{m}=\sqrt{\frac{\sigma_{m}}{\pi}}$

.

(15)

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The duration of the filament $m$ appearing in the time series is given by$\Delta t_{m}=\frac{2_{\beta m}}{v}$

.

(16)

in
and the pulse width
When we choose a filament $m$ , we have the pulse height
$m$
to
the
proportional
the graph of $j(t)$ . The probability of choosing the filament must be. The long-term average of the total duration of the filament $m$“cross section”, which is consistent with (14).
appearing in $j(t)$ is given by$\Delta t_{m}$$J_{m}$$\rho_{m}\propto\sigma_{m}^{1/2}$$\rho_{m}\Delta t_{m}\propto\sigma_{m}\propto q_{m}$4.0–$\beta=1$$\beta=1\sigma----$$3.0$$\frac{\wedge}{\mathrm{v}}$$\sim$

2.0$.\wedge\vee.\neg$

1.0

$—–$

2.0
time [msl

1.0

0.0

$-$

$-$

4.0

3.0

T. Suzuki and Z. Yoshida, Fig. 3

Figure 3: Time series $j(t)$ produced by the statistical$\mathrm{m}o$del ( $\beta=1$ and $\beta=10^{5}$ ).

Controllable parameters in the present statistical model are the velocity , the number, and the inverse temperature . Byof filaments $N$ , the “energy level”
choosing an appropriate set of parameters, we can reproduce the spectral structure of theexperimentally observed fluctuations of the current density. In Fig. 3, we show an example$N=100$ andof the time series $j(t)$ generated by the model with assuming$\beta=10^{5}$. Here,$\beta=1$ .
We also show the non-fluctuating time series given by$(0, J_{\max}]$ .
We
normalize,thetointervaloverrespectwithuniformly,
distributes
gives the mean current density. The typical value ofby imposing that (J). To compare with the experiment (Fig. 1),
is about 1/200 of the cross section
Theaverage radius of a filament (defined by
to
0.
.
be
of
radiuswe take the.
is 1.67
Figure 4 shows the frequency spectrum of $j(t)$ in Fig. 3, in comparison with the experimental spectrum corresponding to Fig. 1. We observe that the theoretical spectrum agrees
well with the experimental one. The spectrum of $j(t)$ has two distinct frequency ranges;
is “white”, while in the highthe spectrum in the low frequency range$v$$\{J_{1}, J_{2}, \cdots, J_{N}\}$$\beta$$v=2\mathrm{k}\mathrm{m}/\mathrm{s},$$\{J_{m}\}$$J_{\max}$$m$$= \sum_{m}J_{m}q_{m}$$|\Omega|$$\sigma_{m}$$|\Omega|$$\rho_{\mathrm{c}}=\Sigma_{m}\rho_{m}J_{m}/\Sigma_{m}J_{m})$$22\mathrm{m}$$\mathrm{x}10^{-2}\mathrm{m}$$(f<f_{c}\approx 20\mathrm{k}\mathrm{H}\mathrm{Z})$227

frequency range $(f>f_{c})$ , the spectrum is the $1/f$ spectrum. In the theory, the criticalfrequency is given by$f_{\mathrm{c}}$$f_{c}= \frac{1}{\pi\Delta i_{c}}=\frac{v}{2\pi\rho_{c}}$whereis the average radius andthe above example,$\rho_{\mathrm{c}}$

,

(17)

is the average duration of a filament; cf. (16). In
and
yield
.

$\triangle t_{\mathrm{c}}$$\rho_{c}=1.67\cross 10^{-2}\mathrm{m}$$v=2\mathrm{k}\mathrm{m}/\mathrm{s}$$f_{c}\simeq 20\mathrm{k}\mathrm{H}\mathrm{z}$T. Suzuki and Z. Yoshida, Fig. 4

Figure 4: Power spectrum of $j(t)$ produced by the statistical model (Fig. 3), in comparison
with the experimental spectrum corresponding to Fig. 1. We apply different normalizations
to both spectra to plot them at different vertical positions. A triangle filter (weighted by
3 point for both sides) is applied.
The two different frequency ranges of the spectrum can be explained as follows. In along time scale
andcome from different randomly-selectedfilaments. Therefore, we obtain a white spectrum. Agreement of our theoretical resultwith the experimental spectrum in this frequency range justifies our assumption of random
movement of filaments.
In the short time scale range, the shape of each pulse (height
and width)
dominates the spectral structure. The Fourier transform of a rectangular pulse yields$(|t_{1}-t_{2}|>\triangle t_{c}),$$j(t_{1})$$j(t_{2})$$=\sqrt m$$F_{m}(f) \propto|\frac{\sqrt m}{f}\sin(\pi\Delta t_{m}f)|$.

$=\Delta t_{m}$(18)Summing up (18) for various pulses “ ”, we obtain the $1/f$ spectrum for the frequency
range of $f>f_{c}$ . Therefore, the $1/f$ spectrum in the high frequency range implies thatthe current density is approximately uniform in each filament, which justifies the model offilaments introduced in the present theory.
$m$

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5

Summary

We have derived a statistical model of filament currents in a plasma, which reproduces thecharacteristic structure of the frequency spectrum of intermittent current fluctuations. Theof filaments under the constraint oftheory gives the Boltzmann distribution of the size
the total current. An interesting assertion is that the time series produced by the model has
is Gaussian.
non-Gaussian power-law spectra, although the statistical distribution ofThis is because the spectrum of the time series is primarily characterized by the probability.
(see (14)) wheregroup for their discussions. This
experiment
The authors are grateful to the REPUTE
work was supported by Grant-in-Aid for Scientific Research from the Japanese Ministry of
Education, Science and Culture No. 09308011.
$\sigma_{m}$$\sigma_{m}$$q_{m}\propto\sigma_{m}$$p_{m}(\sigma_{m})\propto\exp(-\beta J_{m}\sigma_{m})$References[1] T. Suzuki, Intermittent Fluctuations of internal Current in a Turbulent Plasma, disertation, University of Tokyo (1999).
A Tentative Dictionary, edited by[2] Y. Couder, S. Douady and O. Cadot,
P. Tabeling and O. Cardoso (Plenum, New York, 1995), p.131.
$\tau_{ur}bulence_{J}$[3] J. O. Stenflo, Sol. Phys. 32, 41 (1973).

[4] V. Angelopoulos et al., J. Geophys. Res. 97, 4027 (1992).

[5] D. Biskamp and H. Walter, Phys. Fluids B1, 1964 (1989); Phys. Fluids B2, 1787
(1990).
[6] Z. Yoshida, Phys. Rev. Lett. 77, 2722 (1996).
[7] E. D. Siggia, J. Fluid Mech. 107, 375 (1981).

[8] The dynamics of a system of current filaments was studied by Y. Yatsuyanagi, T. Hatori and T. Kato, J. Phys. Soc. Jpn. 65, 745 (1996); see also D. W. Moore andP. G. Saffman, Philos. Trans. R. Soc. Lond. Ser. A 272, 403 (1972); R. Kinney,T. Tajima, J. C. McWilliams and N. Petviashvili, Phys. Plasmas 1, 260 (1994).
[9] J. C. Ingraham et al., Phys. Fluids B2,143 (1990).
[10] A plasma with imperfect magnetic surfaces must have a microscopically filamented
current structure; see J. B. Taylor, Phys. Fluids B5, 4378 (1993).

[11] Z. Yoshida et al., J. Plasma Phys. 59, 103 (1998).
[12] J. B. Taylor, Phys. Rev. Lett. 33, 1139 (1974).