Ion Stochastic Heating by Low-frequency Alfvén Wave Spectrum

TL;DR

This work investigates ion heating by finite-amplitude, low-frequency Alfvén wave spectra, identifying a chaos-based criterion via the effective relative curvature radius $P_{eff}$ that signals breakdown of magnetic moment conservation and onset of stochastic heating. By solving the multi-mode ion equations of motion in a wave field and employing gradient-descent methods to locate the minimum chaotic threshold $P_{eff}^m$, the authors show that broader wave spectra markedly lower the chaos border and drive heating across a wide parameter space. They derive a dimensionless heating-rate scaling $\tilde{Q} = H(\alpha) \tilde{v}^3 \tilde{B}_w^2 \tilde{\omega}_1$, with $H(\alpha)$ peaking near $\alpha \approx 80^\circ$, and qualitatively explain anisotropy using a uniform solid-angle diffusion model on the velocity sphere. The results imply substantial stochastic heating in the solar wind and corona, provide a practical heating-rate formula for quasi-perpendicular AW turbulence, and suggest a joint mechanism linking stochastic heating to wave-driven instabilities.

Abstract

Finite-amplitude low-frequency Alfvén waves are commonly found in plasma environments, such as space plasmas, and play a crucial role in ion heating. The nonlinear interaction between oblique Alfvén wave spectra and ions has been studied. As the number of wave modes increases, ions are more likely to exhibit chaotic motion and experience stochastic heating. The stochastic heating threshold in the parameter space can be characterized by a single parameter, the effective relative curvature radius $P_{eff.}$. The results show excellent agreement with the chaotic regions identified through test particle simulations. The anisotropic characteristics of stochastic heating are explained using a uniform solid angle distribution model. The stochastic heating rate $Q=\dot{T}$ is calculated, and its relationship with wave conditions is expressed as $Q/(Ω_i m_i v_A^2) = H(α) \tilde{v}^3 \tilde{B}_w^2 \tildeω_1$, where $α$ is propagating angle, $Ω_i$ is the gyrofrequency, $m_i$ is the ion mass, $v_A$ is the Alfvén speed, $\tilde{v}$ is the dimensionless speed, $\tilde{B}_w$ is the dimensionless wave amplitude, and $\tildeω_1$ is the lowest dimensionless wave frequency.

Figures (11)

• Figure 1: Time series of chaotic motion under wave conditions $B_w^2/B_0^2=0.15,\,\omega_1/\Omega_i=0.05,\,\tan\alpha=10$ and $N=10$. The particle's initial state is $(0,\,...,\,0,\,0,\,0,\,-1)$. (a) $\mu_m^*$. (b) Magnitude of the magnetic field $|\bm B|/B_0$, a mirror-like field is marked. (c) Parallel velocity ${v_{\parallel}}=\bm{v}\cdot{\langle\bm{B}\rangle_{\Omega_i}}/{\left|\langle\bm{B}\rangle_{\Omega_i}\right|}$, the dashed line indicates the positions where velocity reverses, i.e., where $v_\parallel=0$. (d) Black line: effective relative curvature radius $P_{eff.}$, with a dashed line indicates $P_{eff.}=25$. Green line: the maximum local Lyapunov exponent $\lambda_{local}$ calculated over one gyro-period.
• Figure 2: (a) The change in $\mu_m^*$ between neighboring gyro-periods $\Delta\mu_m^*=\frac{1}{2}\left(\left|\mu_m^*-\mu_{m,-1}^*\right|+\left|\mu_m^*-\mu_{m,+1}^*\right|\right)$ at different $P_{eff.}$, where $\mu_{m,-1}^*$ and $\mu_{m,+1}^*$ represent the values of $\mu_m^*$ of the previous and next gyro-periods, respectively. Different colors represent particles with different initial states. A total of 50 particles are considered, each with speed $v=v_A$, initial pitch-angles $\theta_0=\angle(\bm{v_0},\,\bm{B_0})$ uniformly distributed in $[0, \pi]$, initial azimuth angle $\phi_0=\arctan(v_{y0}/v_{x0})=0$, and initial phases $\psi_{k0}$ uniformly distributed in $[0, 2\pi]$. The dashed line indicates $P_{eff.}=25$. Wave conditions are the same as those in Fig. \ref{['fig:singleParticleTimeSeries']}. (b) The particle's trajectory (black line) and magnetic field lines (red and blue lines) at the period corresponding to the mirror-like field marked in Fig. \ref{['fig:singleParticleTimeSeries']}(b). The particle start from the positions marked by the green dot. The 2 light blue arrows mark the position where $\mu_m^*$ changes.
• Figure 3: $P_{eff.}^m$ varies with wave conditions: (a) $N$ (b) $B_w^2/B_0^2$ (c) $\omega_1/\Omega_i$ (d) $\tan\alpha$.
• Figure 4: $P_{eff.}^m$ and $CR$ in the parameter space $(\omega_1/\Omega_i,\,B_w^2/B_0^2)$, $\tan\alpha=5$. (a)-(d) $P_{eff.}^m$ in the parameter space, with $N=1,\,2,\,5,\,15$. (e)-(h) $CR$ and contour lines of $CR$ and $P_{eff.}^m$ in the parameter space for $N=1,\,2,\,5,\,15$. The $CR$ calculation considered $2500$ particles with $v=v_A$, $\theta_0$ uniformly distributed in $[0, \,\pi]$, $\phi_0 = 0$, and $\psi_{k0}$ randomly distributed in $[0,\, 2\pi]$. The blue lines represent the $CR = 0.01$ contour, while the red lines correspond to the $P_{eff}^m = 25$ contour.
• Figure 5: Ion temperature $T$ varies over time with the parameters $B_w^2/B_0^2=0.15$, $\omega_1/\Omega_i=0.1$, $\tan\alpha=5$. The vertical dashed lines indicate the time $t = 2\pi/\Omega_i$. Solid lines in different colors correspond to different $N$. (a) Cold plasma, $\beta = 0$. The black solid line represents the heating rate $Q=\dot{T}$ for $N=10$, which is calculated from data within the time interval $[2\pi/\Omega_i,\,100/\Omega_i]$. (b) Low-$\beta$ plasma, $\beta = 0.01$.