Nonlinear Monte Carlo model of superdiffusive shock acceleration with magnetic field amplification

Fast collisionless shocks in cosmic plasmas convert their kinetic energy flow into the hot downstream thermal plasma with a substantial fraction of energy going into a broad spectrum of superthermal charged particles and magnetic fluctuations. The superthermal particles can penetrate into the shock upstream region producing an extended shock precursor. The cold upstream plasma flow is decelerated by the force provided by the superthermal particle pressure gradient. In high Mach number collisionless shocks, efficient particle acceleration is likely coupled with turbulent magnetic field amplification (MFA) generated by the anisotropic distribution of accelerated particles. This anisotropy is determined by fast particle transport, making the problem strongly nonlinear and multiscale. Here, we present a nonlinear Monte Carlo model of collisionless shock structure with superdiffusive propagation of high-energy Fermi accelerated particles coupled to particle acceleration and MFA, which affords a consistent description of strong shocks. A distinctive feature of the Monte Carlo technique is that it includes the full angular anisotropy of the particle distribution at all precursor positions. The model reveals that the superdiffusive transport of energetic particles (i.e., Lévy-walk propagation) generates a strong quadruple anisotropy in the precursor particle distribution. The resultant pressure anisotropy of the high-energy particles produces a nonresonant mirror-type instability that amplifies compressible wave modes with wavelengths longer than the gyroradii of the highest-energy protons produced by the shock.

Figure 1

The mean square displacement $\langle \mathrm{\Delta }{z}^2\rangle$ as a function of time for a particle that is propagating with a probability distribution of scattering lengths as given by Eq. (17). The displacement is the projection on one coordinate for a full 3D propagation. The solid (red) line is the Monte Carlo result for $\nu =2.1$, which demonstrates $\langle \mathrm{\Delta }{z}^2\rangle =A_D{t}^b$ with $b\simeq 1.76$, i.e., superdiffusive propagation. The solid (blue) line is the result for $\nu =4.5$. This is hardly distinguishable from the pure diffusion case (where $b=1$) shown by a dotted curve. The dashed (black) curve shows ballistic motion where $b=2$. The curves are normalized to $\langle \mathrm{\Delta }{z}^2\rangle$ at $t=50t_0$, where $t_0={\lambda }_0/v_{\mathrm{pf}}$, and $v_{\mathrm{pf}}$ is the particle speed in the plasma frame. These consistency checks were done using $A_{\theta }^2={\pi }^2/2$ in the Monte Carlo algorithm. For the $\nu =2.1$ and 4.5 results, ${10}^6$ Monte Carlo particles of the same energy were propagated in a uniform upstream flow in order to check our Lévy-walk algorithm.

Figure 2

In all panels, the dashed (red) curves show the results for an unmodified shock with $R_{\mathrm{tot}}=4$. The solid (black) curves show self-consistent results where the momentum and energy fluxes are conserved across the shock including the total escaping energy flux $q_{\mathrm{esc}}$, i.e., $Q_{\mathrm{esc}}\left(p\right)$ summed over $p$. For this example, where all four instabilities are active, the self-consistent compression ratio is $R_{\mathrm{tot}}\simeq 7.2$, and $\sim 20\%$ of the energy flux is lost at the FEB at $z=-{10}^8{r}_{g0}\sim -0.56$ pc. The subshock with $R_{\mathrm{sub}}\sim 3$ is indicated in the upper right-hand panel. All quantities are scaled to far upstream values and note the split log-linear $x$ axis.

Figure 3

Shown are proton phase-space distributions measured in the shock rest frame. Downstream spectra are plotted, as are distributions of particles escaping at the upstream FEB [i.e., Eq. (22)], as indicated. Spectra for unmodified (UM) shocks are in the top panel, while those for consistent nonlinear (NL) shocks are in the bottom panel and the inset. All spectra are absolutely normalized relative to each other. In the Lévy-flight examples (solid and dashed black curves), superdiffusive EP propagation occurs between the FEB at $z_{\mathrm{FEB}}=-{10}^8{r}_{g0}\sim 0.56$ pc and $z_{\mathrm{LF}}=-{10}^4{r}_{g0}$. The normal diffusion cases are shown with dot-dashed red curves and dotted blue curves. We have included turbulence cascade in the nonlinear cases. For the unmodified shocks, the scattering is uniform without magnetic field growth.

Figure 4

Same as the bottom panel in Fig. 3 without the turbulence cascade. In the nonlinear (NL) Lévy-flight examples (solid and dashed black curves), superdiffusive EP propagation occurs between the FEB at $z_{\mathrm{FEB}}=-{10}^8{r}_{g0}\sim 0.56$ pc and $z_{\mathrm{LF}}=-{10}^4{r}_{g0}$. The normal diffusion cases are shown with dot-dashed red curves and dotted blue curves.

Figure 5

Proton anisotropies, as defined in Eqs. (24) and (26), in the rest frame of the background plasma. The top panel, for unmodified (UM) shocks, shows examples in which superdiffusion with $\nu =2.1$ occurs between the FEB and the upstream position $z_{\mathrm{LF}}=-{10}^4{r}_{g0}$ (black dashed curves), $z_{\mathrm{LF}}=-{10}^5{r}_{g0}$ (green dot-dashed curves), and $z_{\mathrm{LF}}=-{10}^6{r}_{g0}$ (blue dotted curves). The bottom panel shows the anisotropy for nonlinear shocks with cascading (blue solid curves) and without (black dashed curves) for $z_{\mathrm{LF}}=-{10}^4{r}_{g0}$. Results with no superdiffusion are shown with cascading (red dotted curves). The fluctuations in the $A_2$ results at large $\left|z\right|$ are statistical errors from the Monte Carlo simulation.

Figure 6

Spectral energy densities of the EP-driven magnetic fluctuations measured at three positions: (a) the FEB, (b) $z=0.01z_{\mathrm{FEB}}$, and (c) in the downstream region. There is a strong effect on $kW\left(k\right)$ from the turbulence cascade. For both of these nonlinear shocks, $z_{\mathrm{LF}}=-{10}^4{r}_{g0}$ and $\nu =2.1$.

Figure 7

The bulk flow speed (top panel) and scattering center velocity derived from the nonlinear Monte Carlo model (bottom panel) with and without the turbulent cascade, as indicated. Also shown in the bottom panel is $v_{\mathrm{alf}}\left(B_{\mathrm{ls}}\right)$, the local Alfvén speed derived using the amplified magnetic field (dashed curves). For these examples, $z_{\mathrm{LF}}=-{10}^4{r}_{g0}$.