Charged particle energization by low-amplitude electrostatic waves at cyclotron harmonics

TL;DR

Problem: energization of ions by low-amplitude electrostatic waves at cyclotron harmonics in magnetized plasmas. Approach: Hamiltonian analysis using action-angle variables and a Jacobi-Anger style expansion, comparing exact resonance and off-resonance with slowly varying wave amplitude. Key contributions: sub-threshold energy transfer is non-adiabatic and yields a final energy that is bounded and determined by the wave-train, enabling energization of cold ions. Significance: the mechanism can contribute to ion heating and thermal content in magnetized plasmas even when individual waves do not satisfy the conventional heating threshold.

Abstract

The system made by a charged particle interacting with a single electrostatic wave which propagates perpendicularly to the magnetic field, at a frequency larger than the cyclotron one, has been extensively studied in literature due to its implications with ion heating in magnetized plasmas. It is known that a threshold in the electrostatic potential must be exceeded in order for stochastic particle motion and heating to occur. Regardless its amplitude, however, the electrostatic wave induces a periodic oscillation in the particle motion. We show, by analytical and numerical arguments, that this dynamics is non-adiabatic, meaning that the particle does not land back to its initial state when the wave is slowly turned off. This way, particle energization (although, not rigorously heating) occurs even under sub-threshold conditions.

Figures (5)

• Figure 1: Contour plot of $J_M (r) \cos(M \theta')$ for $M = 2$. The red and green curves are respectively the solutions of the truncated and full Hamiltonian equations. Throughout this paper, by full Hamiltonian we will mean that retain all terms within the interval $(M-7, M+7)$. The starting position is marked by the black dot. The wave amplitude is $A = 1/3$.
• Figure 2: Time traces of $I(t)$ computed using the full Hamiltonian (\ref{['eq:hf']}) with $M = 2$. The blue, orange and green curves correspond to $I(0)$ as given in the legend, while the initial phase is $\theta'(0) = \pi/8$ for all. The initial conditions, thus, have been chosen close to the bottom of the ellipsoidal curve tracked by the green trajectory in Figure (1). The wave amplitude is $A = 1/3 f(t)$, with $f(t)$ defined in Eq. (\ref{['eq:shape']}), and plotted in the figure as a red curve. Its parameters are $dt = 20, t_s = 100, t_f = 400$. Notice that, with this choice of parameters, the time needed for $A$ to grow from zero to its flattop value is $\approx 130$ time units, equivalent to about 20 Larmor periods.
• Figure 3: This figure has the same content of Figure (\ref{['fig:due']}). In this case, the initial conditions are the same for all trajectories $( I(0) = 1, \theta'(0) = \pi/8)$, but the shape function grows over very slightly different time scales between runs: $t_s/dt = 5,5.25, 5.5$ for the three cases. The yellow rectangle centered at $t \approx 100, I \approx 1)$ marks the region where $f$ differs between runs. All other parameters are the same from the previous figure.
• Figure 4: Contour plots of $H'$ (Eq. \ref{['eq:ptr']}) for different combinations of the parameters $\Delta M, M, A$.
• Figure 5: Examples of trajectories for different sets of parameters and different initial conditions. In all subplots, the initial conditions are the same: $I(0) = 3, \theta(0) = -1.3$ (blue curve); $I(0) = 8, \theta(0) = -1.0$ (orange curve); $I(0) = 8, \theta(0) = 0.1$ (green curve). The red curve is the shape function $f$ modulating the amplitude $A$, like in Fig. (\ref{['fig:due']}). The parameters $M, \Delta M, A$ employed for each subplot are reported as titles of the subplot itself. All numerical trajectories are computed using the full hamiltonian (\ref{['eq:hf']}), not their truncated versions.