High-beta equilibrium in mirror machine with population of fast sloshing ions

TL;DR

The paper addresses constructing axisymmetric high-$\beta$ equilibria in mirror machines with a population of fast sloshing ions produced by off-axis neutral beam injection. It introduces an adiabatic-invariant framework for betatron-moving ions, defines a longitudinal velocity invariant $V_z$, and uses a composite fast-ion distribution to generate pressure and current that feed a Green's-function-based Grad-Shafranov solution. Numerical results demonstrate a self-consistent, unique high-$\beta$ equilibrium and verify adiabatic invariance for the majority of fast ions via Poincaré mappings, even when the axis field weakens. The work supports the feasibility of diamagnetic, high-$\beta$ mirror configurations sustained by off-axis fast ions and provides a methodology to diagnose pressure profiles and stability implications in such regimes.

Abstract

A method of constructing the high-beta (diamagnetic-bubble-like) equilibrium with a population of fast sloshing ions is discussed. Fast ions move along betatron orbits; such ions can arise because of off-axis neutral beam injection. Conservation of the adiabatic invariant of these ions is proposed; a simplified expression for the invariant is presented. Numerical examples of equilibrium with sloshing ions are shown and conservation of the invariant is justified by a direct numerical simulation of motion of fast ions even in the case with beta equal to 1.

Figures (6)

• Figure 1: Examples of longitudinal distribution of magnitude of magnetic field at axis. Flat (left) and peaked (right) distributions with $n_{\max}=25$. Full azimuthal current $J_\theta$ varies from zero to 270 kA with step of 24.5 kA.
• Figure 2: Examples of radial distribution of $z$-component of magnetic field at $z=0$ for flat (left) and peaked (right) distributions. Parameters are the same as in figure \ref{['Bz']}.
• Figure 3: Examples of field lines in case of flat distribution with $n_{\max}=25$ and $J_\theta=270$ kA.
• Figure 4: Dependence of magnetic field magnitude in center in case of flat (upper points) and peaked (lower points) distributions. Zero approximation: vacuum flux (squares) and magnetic field for $J_\theta=270$ kA (circles).
• Figure 5: Examples of Poincare mapping for ions with azimuthal momentum $p_\theta/(m\Omega_v)=-39$ cm$^2$ and kinetic energy $\sqrt{2\varepsilon/(m\Omega_v^2)}=11$ cm (upper), $\sqrt{2\varepsilon/(m\Omega_v^2)}=12.5$ cm (middle) and $\sqrt{2\varepsilon/(m\Omega_v^2)}=13.5$ (down). Magnetic field are the same as in figure \ref{['FieldLines']}.