Collisionless particle dynamic in an axi-symmetric diamagnetic trap

TL;DR

This work analyzes collisionless single-particle dynamics in an axisymmetric diamagnetic trap at high beta, where the magnetic moment $m v_\perp^2/(2B)$ can fail to be conserved due to diamagnetic field extrusion. Using a Hamiltonian framework with $H=\frac{p_r^2}{2m}+\frac{(p_\theta-e\Psi/c)^2}{2mr^2}+\frac{p_z^2}{2m}+e\varphi$, the paper identifies two unlimited-confinement mechanisms—absolute confinement (for $\Omega p_\theta<0$ and $\varepsilon>-\,R_v\Omega p_\theta$) and adiabatic confinement via the radial invariant $I_r$ (when the field varies smoothly). It then analyzes how corrugation of the vacuum field can destroy adiabaticity by inducing resonances, deriving a Chirikov-type criterion with a quantitative adiabaticity bound $8\delta b k a I_0(ka)\frac{v_0^2}{v_\perp^2}\left(1-\frac{r_{\min}^2}{a^2}\frac{2v_0^2-v_\perp^2}{v_\perp^2}\right)<1$, and provides lifetime estimates for chaotic and gas-dynamic regimes. Numerical examples illustrate how bubble geometry and corrugation affect confinement, showing that kinetic effects can substantially extend confinement times beyond simple gas-dynamic predictions. Overall, the results highlight the importance of adiabatic invariants and field structure for diamagnetic-trap confinement and offer guidance for kinetic modeling of high-beta mirror devices.

Abstract

Particle dynamic in an axi-symmetric mirror machine with an extremely high plasma pressure equal to pressure of vacuum magnetic field (so-called regime of diamagnetic confinement) is investigated. Extrusion of magnetic field from central region due to plasma diamagnetism leads to non-conservation of the magnetic moment and can result in chaotic movement and fast losses of particles. The following mechanisms can provide particle confinement for unlimited time: absolute confinement of particles with high azimuthal velocity and conservation of adiabatic invariant for particle moving in smooth magnetic field. The criteria of particle confinement and estimations of lifetime of unconfined particles are obtained and verified in direct numerical simulation. Particle confinement time in the diamagnetic trap in regime of gas-dynamic outflow is discussed.

Figures (5)

• Figure 1: An example of trajectories of particles with $\Omega p_\theta<0$ (left), $\Omega p_\theta=0$ (center) and $\Omega p_\theta>0$ (right). Dashed circle bounds region $r<a$ where magnetic field is less than $B_0/2$. Arrows indicate direction of mean azimuthal velocity.
• Figure 2: An example of dependence of plasma density $n/n_{i0}$ (red) and magnetic field $B_z/B_0$ (blue) on radius in trap center. Parameters: $(2T_i/(m_i\Omega^2))^{1/2}=2$, $a=20$, vacuum mirror ratio $R_v=2$.
• Figure 3: An examples of magnet coils (rectangles) and magnetic field line on "bubble" boundary (solid curve) for smooth (left) and corrugated (right) vacuum magnetic field. Parameters: $(2T_i/(m_i\Omega^2))^{1/2}=2$, $a=20$.
• Figure 4: An example of value of critical longitudinal velocity for ions with $\rho=2$ moving in corrugated magnetic field (points) and margin of adiabaticity (\ref{['corS07']}) at $ka=2.7$ and $\delta b=0.01$ (dashed line).
• Figure 5: An example of number of unconfined ions after $n$ bounce oscillations (points) and function $19e^{-x/18.3}$ (solid). Parameters: $\rho=2$, $r_{\min}=0$.