Chaotic Motion of Ions In Finite-amplitude Low-frequency Alfvén Waves

TL;DR

This work demonstrates that ions in finite-amplitude, low-frequency Alfvén waves can experience chaotic motion through WFLC-induced pitch-angle scattering. By analyzing the maximum Lyapunov exponent $\lambda_m$ and Chaos Ratio $CR$ for ensembles of ion initial states, the authors identify an onset threshold linked to the effective relative curvature radius $P_{eff}$, with a critical value around $C\approx 25$. An analytic criterion $P_{eff}^m(B_w^*,\mathbf{k^*}) = \frac{1}{k_z^* B_w^* \sin\alpha}(1+B_w^{*2}-2 B_w^* \sin\alpha)^{3/2} < C$ accurately maps the global chaos region in parameter space and aligns with the $CR=0.01$ boundary. The study highlights WFLC as a fundamental mechanism for stochastic heating in Alfvénic plasmas, with potential relevance to solar wind, corona, and magnetospheric dynamics, and provides data/code access for reproducibility.

Abstract

Finite-amplitude low-frequency Alfvén waves (AWs) are commonly found in plasma environments, such as space plasmas, and play a crucial role in ion heating. In this study, we examine the nonlinear interactions between monochromatic AWs and ions. When the wave amplitude and propagation angle lie within certain ranges, particle motion becomes chaotic. We quantify this chaotic behavior using the maximum Lyapunov exponent, $λ_m$, and find that chaos depends on the particles' initial states. To characterize the proportion of chaotic particles across different initial states, we introduce the Chaos Ratio ($CR$). The threshold for the onset of global chaos is calculated as the contour line of $CR=0.01$. We analyze changes in the magnetic moment during particle motion and identify the physical image of chaos as pitch-angle scattering caused by wave-induced field line curvature (WFLC). Consequently, the condition for chaos can be expressed as the effective relative curvature radius $P_{eff.}<C$, with $C$ being a constant. We analytically determine the chaos region in the $(k_x,\,k_z,\,B_w)$ parameter space, and the results show excellent agreement with the global chaos threshold given by $CR=0.01$.

Figures (4)

• Figure 1: Chaos quantification indices. (a) $\lambda_{m}$ in the $k_x^*-B_w^*$ parameter space, with $k_z^* = 0.25$. The initial state is $(0,\,0,\,0,\,-1)$. (b) $\lambda_{m}$ for different values of $B_w^*$ and initial phases $\psi_0$, with the particle's initial velocity set to $(0,\,0,\,-v_A)$, $k_x^* = 0.5$, and $k_z^* = 0.25$. The red line is the Chaos Ratio $CR$ among all $1000$ initial states, where particle motion is classified as chaotic when $\lambda_{m}>0.0025$.
• Figure 2: Time series of chaotic motion with parameters $B_w^*=0.5,\,k_x^*=0.5,\,k_z^*=0.05$. The initial state of the particle is $(0,\,0,\,0,\,-1)$. (a) Parallel velocity ${v_{\parallel}}$, the dashed line indicates the positions of velocity reversal, i.e. $v_\parallel=0$. (b) Magnitude of the magnetic field $|\bm B|/B_0$. (c) $\mu_m^*$, the arrow marks the position of the $\mu_m^*$ change plotted in Fig. \ref{['fig:muChange']}(b). (d) Black line: effective relative curvature radius $P_{eff.}$, the dashed line indicates $P_{eff.}=25$. Green line: the maximum local Lyapunov exponent $\lambda_{local}$ during one gyro-period.
• Figure 3: (a) Changes in the magnetic moment $\Delta\mu_m^*$ at different values of $P_{eff.}$, with different colors representing particles with different initial states. 50 particles are considered, each with speed $v=v_A$, initial pitch-angles $\theta_0$ uniformly distributed in $[0, \pi]$, initial azimuth angle $\phi_0=0$, and initial phases $\psi_0$ uniformly distributed in $[0, 2\pi]$. The dashed line indicates $P_{eff.}=25$. Parameters are the same as Fig. \ref{['fig:singleParticleTimeSeries']}. (b) The particle's trajectory (black solid line) and magnetic field lines at the time corresponding to the arrow in Fig. \ref{['fig:singleParticleTimeSeries']}(c). The red line represents the field line particle gyrating around before the $\mu_m^*$ change, while the blue line represents the field line particle gyrating around after the $\mu_m^*$ change. The 4 black dashed lines indicate trajectories with slight differences in initial velocities. (c) Trajectories of 9 initially adjacent particles over 100 gyro-periods (black lines). The particles start from the positions marked by the green dot. Two magnetic field lines are plotted for reference (red lines). Positions where $P_{eff.}<25$ are marked in blue; pitch-angle scattering events that cause particle trajectory separation are marked with 3 light blue arrows; and positions where $|v_{\parallel}| < 0.05$ are marked in yellow, indicating the mirror points of the bouncing motion.
• Figure 4: $CR$ and contour lines of $CR$ and $P_{eff.}^m$ in $B_w^*-k_x^*$ parameter space for $k_z^* = 0.1$, $0.25$, and $0.5$. The $CR$ calculation considered $4000$ particles with $v=v_A$, $\theta_0$ uniformly distributed in $[0, \,\pi]$, $\phi_0 = 0$, and $\psi_0$ uniformly distributed in $[0,\, 2\pi]$. The blue lines in (a), (b), and (c) represent the $CR = 0.01$ contour, while the red lines correspond to the $P_{eff}^m = 25$ contour. (d) $CR = 0.01$ contour lines for $k_z^* = 0.1$, $0.25$, and $0.5$.