Admissible Equilibria of Non-neutral Plasmas in a Malmberg-Penning Trap

A “parallel current constraint” is derived, that in combination with the Poisson equation allows one to select admissible equilibria of non-neutral plasmas in a Malmberg-Penning trap in the presence of a nonuniform and nonaxisymmetric magnetic field. Asymmetry-induced currents (analogous to the Pfirsch-Schlüter currents in Tokamaks) appearing in a non-neutral plasma even in the absence of magnetic drifts are explicitly computed in the case of a uniformly tilted magnetic field.

Figure 1

Unperturbed density $n_0$ (solid line), unperturbed electric potential ${\varphi }_0$ (dashed line), and radial part ${\varphi }_1^{(1)}$ (dot-dashed line) of the perturbed potential used for calculating the Pfirsch-Schlüter currents vs the flux radius $\rho =(2\psi /B_{*}{)}^{1/2}$ normalized over the radius $R$ of the MP trap. The density is normalized by its maximal value $n_{*}$, and the potentials by $(T/e)(a/{\lambda }_D{)}^2$, with ${\lambda }_D\equiv [T/(4\pi e^2{n}_{*}){]}^{1/2}$ the Debye length. The density profiles corresponds to a global thermal equilibrium with a column radius $a/R=0.25$ (computed at $1/2$ of the maximal density), and ${\lambda }_D/R=0.05$.

Figure 2

Level curves of ${\Psi }_1$ (streamlines of the asymmetry-induced current density) on a flux surface with $\rho /R=0.25$ for a uniform magnetic field with a tilt angle $\alpha =1°$. Solid and dashed lines correspond to a clockwise and a counterclockwise flowing current, respectively. The total length of the plasma column is $L=8R$. Other parameters are indicated in Fig. 1.

Figure 3

Radial profiles of the parallel current density $j_{\parallel }$ for a uniform magnetic tilt along the ray $\vartheta =\pi /2$, $\chi =0$ (midplane of the trap) for $a/R=0.25$ (solid lines), $a/R=0.5$ (dashed lines) and various values of ${\lambda }_D/R$ (indicated on the plot); $j_{\parallel }$ is normalized by $(\alpha c{n}_{*}T/B_{*}{\lambda }_D)(L/R{)}^2(a/{\lambda }_D{)}^2$.