Magnetically confined charged particles: From steep density profiles to the breaking of the adiabatic invariant

Abstract

This study examines the stability of Vlasov equilibrium solutions for magnetically confined plasmas, derived through the principle of maximum entropy. By treating the toroidal limit as a perturbation from an analytical cylindrical solution, we demonstrate that these equilibria align well with the inviscid magnetohydrodynamic (MHD) description. Using the aspect ratio as a perturbation parameter, we compute particle trajectories sampled from the kinetic equilibrium distribution, confirming the overall stability of the solutions. However, under burning plasma conditions, chaotic dynamics emerge for particles with supra-thermal and even thermal energies. This destroys the adiabatic invariance of the magnetic moment. The exact consequences are unclear, but they could undermine the foundational assumptions of gyrokinetic modelling in burning plasmas. Nevertheless, these results suggest the possibility of unaccounted transport losses in future burning plasma operations. The interplay between turbulence and energetic particles in the presence of Hamiltonian chaos certainly warrants further investigation.

Figures (4)

• Figure 1: Notations used in the paper. In the infinite aspect ratio limit ($R\rightarrow\infty$), the torus locally becomes a cylinder, the angle $\varphi$ matching $\frac{z}{R}$. A rotational invariance around the $\theta$ direction is recovered, and the rotational invariance around $\varphi$ becomes a translational one along $z$ with $2\pi R$ periodic boundary conditions
• Figure 2: Stability of radial density in toroidal geometry for $R=100~\mathrm{m}$, $1/\beta=10~\mathrm{keV}$, $B_{0}=1~\mathrm{T}$, $a=430$ and $b=-13$. The simulation was performed with $2^{20}$ particules, with a time step $\Delta t=0.025~\omega_{c,0}^{-1}$.
• Figure 3: Dynamics of two neighboring particles obtained from $p_{\theta}\simeq0.00175$ and $p_{z}\simeq-0.03453$. Top figure: Poincaré sections, for $R=10^{5},10^{4},10^{3},10^{2}$. On the right part of the plot we see the evolution of the stochastic layer for a particle with an energy $E\simeq10.00~\mathrm{keV}$. On the left is the evolution of a quasiperiodic trajectory without chaos for a lower energy, $E\simeq9.97~\mathrm{keV}$. The trajectories are followed for $t=4\times10^{7}\,\omega_{c,0}^{-1}$, with $\Delta t=0.01~\omega_{c,0}^{-1}$. Bottom figure: Three-dimensional plots of part of the same trajectories in dashed red (quasiperiodic), in dashed blue (chaotic), for $R = 100$.
• Figure 4: Magnetic moment for $R=100$ of the two neighboring trajectories depicted in -evolution-of-section (from $p_{\theta}\simeq0.00175$ and $p_{z}\simeq-0.03453$), and computed with $\Delta t=0.01~\omega_{c,0}^{-1}$. Top figure: Relative fluctuations in the magnetic momentum averaged over a finite time period of $500\,\omega_{c,0}^{-1}$. Central panel: Fourier spectrum of $\mu$ for the quasiperiodic trajectory (at $E\simeq9.97~\mathrm{keV}$). Bottom panel: Fourier spectrum of $\mu$ for the chaotic trajectory (at $E\simeq10.00~\mathrm{keV}$).