Relaxation of Coaxial Nonneutral Magnetized Plasmas

From the source

1. A 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

2. A 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

3. The 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

4. The 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

5. At 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

6. What 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

The way in

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