STM-D-0957Paper2003Published and peer-reviewed

Relaxation of Coaxial Nonneutral Magnetized Plasmas

Kimitaka Itoh · Sanae-I. Itoh · Shinichiro Toda · Nobumitsu Yokoi · Akira Yoshizawa

Abstract and summary · read the original at the source · none found

In one page

Put a hollow tube of electrons — no ions, just electrons — inside a strong magnetic field pointing along its axis, and it spins. Its own charge creates an outward electric field, that field drives a rotation, and because the inner and outer edges of the tube go round at different rates the shear tears it apart, the way wind shear curls the edge of a cloud. Kimitaka Itoh and colleagues at Japan’s National Institute for Fusion Science ask what shape the plasma settles into once that has happened, and answer with a variational principle. Enstrophy — the bookkeeping measure of how much swirl a flow contains — drains away faster than energy does, so the plasma should end up in the lowest-enstrophy shape still available to it at the energy and particle number it started with. The answer comes out as a Bessel function: a smooth column, densest on the axis. Better still, it comes out as one short formula for the column’s width that an experiment can check.

Why it matters hereChapter 9 is about plasma that organises itself into a persistent object instead of dispersing, and this is that process reduced to its simplest possible case — one species, two dimensions, a trap you can build on a bench and watch through a window. Chapter 12 cares because the same relaxation logic, minimise one quantity while another is conserved, is what produces the Taylor state and the spheromak that fusion machines are built from.

What it claims

01A cloud of pure electrons held in a strong axial magnetic field behaves like a two-dimensional fluid. Its own space charge sets up a radial electric field, and that field crossed with the magnetic field makes the whole column rotate azimuthally at a speed equal to the electric field divided by the magnetic field strength. Because the magnetic field is uniform, the rotation speed is simply proportional to the local electric field, which means the electron density plays the part that vorticity plays in an ordinary fluid — and that correspondence is why a trap full of electrons is such a precise laboratory for two-dimensional turbulence.Section 2, equations 2 to 4; the correspondence is the subject of the Dubin and O’Neil review cited as reference 10
Settled physics
02A hollow shell of electrons is unstable, and the instability is the ordinary Kelvin-Helmholtz one. The authors take as their initial condition a cylindrical shell of uniform density between an inner radius and an outer radius, calculate the electric field it generates, and note that the resulting azimuthal velocity is strongly sheared across the shell. That shear drives large-scale Kelvin-Helmholtz instability, the azimuthal symmetry is lost, the plasma is violently deformed and the particles mix — and what emerges afterwards is a single diffused column.Section 1, closing paragraph; Section 2, equations 1 and 3, and Figure 1
Settled physics
03The relaxed state is the minimum-enstrophy state at fixed energy and fixed particle number. Enstrophy decays faster than field energy in large-scale turbulent motion, so the authors minimise the enstrophy integral while holding the electrostatic energy and the total number of electrons per unit length constant, and show in an appendix that conserving particle number already conserves the azimuthal angular momentum, so angular momentum is not a separate constraint. Solving the resulting Euler equation gives the profile in closed form: the radial electric field follows a first-order Bessel function, the density follows a zeroth-order Bessel function, and the plasma edge sits at the first zero of that Bessel function.Section 2, equations 5 to 14; Appendix, equations A1 to A5
Published and peer-reviewed
04The paper’s deliverable is a testable formula for how wide the relaxed column is. Matching particle number and energy between the initial shell and the final Bessel profile gives the final radius as the initial outer radius multiplied by an exponential factor that depends on nothing but the ratio of the two shell radii, through a purely geometrical quantity the authors call F, which lies between zero and one quarter. The column therefore always ends up wider than the shell it came from — between about 1.28 and 1.65 times the original outer radius — and where in that narrow range it lands is fixed by the starting geometry alone, not by the density or the magnetic field. Plotting the predicted radius against the shell ratio is the measurement the paper is asking for.Section 3, equations 16 to 27, especially equations 24, 25 and 26; Figures 2 and 3
Published and peer-reviewed
05At high density the answer shifts, and it shifts more for ions than for electrons. The clean Bessel result assumes the plasma frequency is far below the cyclotron frequency, so the perpendicular dielectric constant of the magnetised plasma is close to one. As the density approaches the Brillouin limit that stops being true, an extra term enters the profile equation, and numerical solution shows the relaxed column becoming narrower: the eigenvalue drops from about 5.783 to about 5.684 and the radius to 0.977 of its low-density value when the squared plasma-to-cyclotron frequency ratio is 0.416. The correction scales with particle mass, so a pure ion plasma shows the same effect far more strongly — which makes singly charged ion traps the easier place to look for it.Section 3, closing discussion, equations 28 and 29; Figure 4
Published and peer-reviewed
06What to watch: which conserved quantity nature actually picks. The authors are explicit that holding the field energy exactly constant is an idealisation — turbulence will dissipate some of it, the final energy will be a little lower than the initial energy, and their radius formula is therefore a lower bound rather than an exact prediction. They also set their result beside two rival variational schemes, one minimising fluid enstrophy under canonical angular momentum and one minimising canonical enstrophy, each of which adds a free parameter and so predicts a family of profiles rather than a single one. Their own result is the tightest special case of both. Measuring the relaxed radius against the shell ratio in a real trap is what separates them.Section 4, Summary and Discussion, paragraphs 2 and 3; references 6 and 12
What to watch

Read it · abstract

Abstract

A variational principle is applied to the relaxation of pure electron plasma in a strong axial magnetic field. The initial cylindrical shell structure of electrons can be unstable against Kelvin Helmholtz instability, and the plasma shape relaxes to its final state having a diffused profile. The shape of the plasma distribution in the final state is given based upon the anzats of the minimum enstrophy, and an experimentally-testable formula is obtained.

Kimitaka Itoh and Shinichiro Toda, National Institute for Fusion Science, Toki; Sanae-I. Itoh, Research Institute for Applied Mechanics, Kyushu University; Nobumitsu Yokoi and Akira Yoshizawa, Institute of Industrial Science, University of Tokyo. Relaxation of Coaxial Nonneutral Magnetized Plasmas, Journal of Plasma and Fusion Research 79, No. 12, pages 1297-1301 (December 2003); received 22 August 2003, accepted 7 October 2003. Published at doi.org/10.1585/jspf.79.1297 and free to read on J-STAGE.

(Abstract only. The paper is free to read at the publisher but carries no reuse licence, so nothing beyond the authors’ own abstract is reproduced here — see the rights note above. The summary and claims were written from the complete five-page paper.)

Companion sheets on this site: the same relaxation logic — minimise one quantity while a slower-decaying one is conserved — is what builds the Taylor state and the spheromak, set out at book length in Paul Bellan’s Magnetic Helicity, Spheromaks, Solar Corona Loops, and Astrophysical Jets and reviewed for the reactor case in Thomas Jarboe’s The spheromak confinement device. For plasma that relaxes into a self-supporting knot rather than a column, see Force-Free Time-Harmonic Plasmoids and Ball lightning as a free-force magnetic knot with linked streamers.

The way in

https://doi.org/10.1585/jspf.79.1297LICENCE CHECKED DIRECTLY. The article is free to read on J-STAGE, which is why the open-access indexes label it bronze, but bronze means only ‘readable at the publisher’ and carries no reuse licence. The J-STAGE record for volume 79, issue 12 states the copyright as ‘2003 by The Japan Society of Plasma Science and Nuclear Fusion Research’ and no Creative Commons statement appears on the article page or anywhere in the PDF, so only the authors’ own abstract is reproduced here. SOURCE READ IN FULL. The complete five-page paper was retrieved from the J-STAGE PDF on 2026-09-08 and read end to end, and every locator below points into that text by section, equation or figure number. NAMES. The journal prints its author line in the Japanese house style, surname first and in capitals — ITOH Kimitaka, ITOH Sanae-I., TODA Shinichiro, YOKOI Nobumitsu and YOSHIZAWA Akira. They are given here given-name-first in normal case. AFFILIATIONS. Kimitaka Itoh and Shinichiro Toda were at the National Institute for Fusion Science, Toki; Sanae-I. Itoh at the Research Institute for Applied Mechanics, Kyushu University; Nobumitsu Yokoi and Akira Yoshizawa at the Institute of Industrial Science, University of Tokyo. EQUATIONS. The PDF extraction flattened superscripts, subscripts and Greek letters and broke several displayed equations across columns; no equation is reproduced here, and where a result is quoted below it is stated in words with the paper’s equation number given as the locator.

How to cite it

Kimitaka Itoh, Sanae-I. Itoh, Shinichiro Toda, Nobumitsu Yokoi, Akira Yoshizawa (2003) Relaxation of Coaxial Nonneutral Magnetized Plasmas. doi:10.1585/jspf.79.1297

Where it sits in the curriculum

Plasmoids, charge clusters and the orbs Lattice confinement fusion