Kinetic Simulations of the Interruption of Large-Amplitude Shear-Alfvén Waves in a High-$\beta$ Plasma

Using two-dimensional hybrid-kinetic simulations, we explore the nonlinear “interruption” of standing and traveling shear-Alfvén waves in collisionless plasmas. Interruption involves a self-generated pressure anisotropy removing the restoring force of a linearly polarized Alfvénic perturbation, and occurs for wave amplitudes $\delta B_{\perp }/B_0\gtrsim {\beta }^{-1/2}$ (where $\beta$ is the ratio of thermal to magnetic pressure). We use highly elongated domains to obtain maximal scale separation between the wave and the ion gyroscale. For standing waves above the amplitude limit, we find that the large-scale magnetic field of the wave decays rapidly. The dynamics are strongly affected by the excitation of oblique firehose modes, which transition into long-lived parallel fluctuations at the ion gyroscale and cause significant particle scattering. Traveling waves are damped more slowly, but are also influenced by small-scale parallel fluctuations created by the decay of firehose modes. Our results demonstrate that collisionless plasmas cannot support linearly polarized Alfvén waves above $\delta B_{\perp }/B_0\sim {\beta }^{-1/2}$. They also provide a vivid illustration of two key aspects of low-collisionality plasma dynamics: (i) the importance of velocity-space instabilities in regulating plasma dynamics at high $\beta$, and (ii) how nonlinear collisionless processes can transfer mechanical energy directly from the largest scales into thermal energy and microscale fluctuations, without the need for a scale-by-scale turbulent cascade.

Figure 1

Out-of-plane magnetic perturbation, $\delta B_z/B_0$, in a standing shear-Alfvén wave at $t=0$ (a), $t=0.08{\tau }_A$ (b), $t=0.3{\tau }_A$ (c), $t=0.45{\tau }_A$ (d), and $t=0.6{\tau }_A$ (e) (${\tau }_A={10}^4{\mathrm{\Omega }}_i^{-1}$ is the linear Alfvén period).

Figure 2

Evolution of the standing wave from Fig. 1: we show the $y$-averaged $\delta B_z/B_0$ (black line, left axis), $\delta u_z/v_A$ (blue dot-dashed line, left axis), and firehose parameter $4\pi \mathrm{\Delta }p/B^2$ (red, right axis; the dotted red line shows $\mathrm{\Delta }p=-B^2/4\pi$), at the times illustrated in Figs. 1, 1, and 1. The background color shows the effective collisionality ${\nu }_c/{\omega }_A$ caused by particle scattering from microscale fluctuations, measured over the time intervals $t/{\tau }_A\in [0.07,0.15]$ (a), $t/{\tau }_A\in [0.2,0.4]$ (b), and $t/{\tau }_A\in [0.55,0.65]$ (c). In (b), we also show (dashed lines) $\delta B_z/B_0$ and $4\pi \mathrm{\Delta }p/B^2$ for a decaying SA standing wave in a Braginskii model at $\beta =100$, ${\nu }_c/{\omega }_A\approx 10$ ($\delta u_z/v_A$ is omitted for clarity).

Figure 3

Plasma heating due to the standing wave in Fig. 1. We compare the rate of change of thermal energy ${\partial }_t{E}_{\mathrm{th}}=\int d\mathit{x}{n}_i\sum _r{\partial }_t({\mathrm{\Pi }}_{rr}/n_i)/2$ (black line; ${\mathrm{\Pi }}_{rs}$ is the pressure tensor), with mechanical heating-$\int d\mathit{x}\sum _{rs}{\mathrm{\Pi }}_{rs}{\nabla }_r{u}_s$ (green dashed line), heating from the large-scale SA wave $\int d\mathit{x}\mathrm{\Delta }\overline{p}{\widehat{b}}_x{\widehat{b}}_z{\partial }_x{\overline{u}}_z$ (blue dot-dashed line; here $\overline{·}$ denotes a filter that smooths fluctuations with $k{\rho }_i\gtrsim 0.25$), and the approximate viscous heating [43] from the SA wave after interruption ${\nu }_c^{-1}\int d\mathit{x}{\overline{p}}_{\parallel }({\widehat{b}}_x{\widehat{b}}_z{\partial }_x{\overline{u}}_z{)}^2$ (red dotted line; we use ${\nu }_c/{\omega }_A\approx 10$ as in Fig. 2). We normalize by $E_{\mathrm{th}}$ and use units of ${\tau }_A$ (note the small rates, due to the high $\beta$). The initial ${\partial }_t{E}_{\mathrm{th}}<0$ is due to the creation of $\mathit{E}$ fluctuations (because of particle noise).

Figure 4

Out-of-plane magnetic perturbation $\delta B_z/B_0$ for a SA traveling wave with ${\lambda }_A=250{\rho }_i$ at $t=0$ (a), $t=0.2{\tau }_A$ (b), $t={\tau }_A$ (c), and $t=3{\tau }_A$ (d).

Figure 5

(a) Scattering rate ${\nu }_c/{\omega }_A$ of the traveling wave in Fig. 4 as a function of $x$ and $t$. The grey lines follow the wave fronts (this is close to where $\mathrm{\Delta }p$ is most negative). (b) Time evolution of $⟨4\pi \mathrm{\Delta }p/B^2⟩$. The shaded region indicates the range of $4\pi \mathrm{\Delta }p/B^2$ seen across the wave profile, to illustrate when the wave can excite firehose modes. (c) Energy of the magnetic perturbation $E_{\delta B}=\int d\mathit{x}\delta B_z^2/8\pi$ (blue) and kinetic energy $E_{\delta u}=\int d\mathit{x}\rho \delta u_z^2/2$ (red), normalized by $E_0=\int d\mathit{x}{B}_0^2/8\pi$. In (b) and (c), we plot the results from an equivalent Landau-fluid simulation [10] (dashed lines), for comparison.