The Importance of Heat Flux in Quasi-Parallel Collisionless Shocks

TL;DR

This work investigates how self-generated non-thermal particles alter collisionless quasi-parallel shock hydrodynamics. Using self-consistent hybrid PIC simulations with the dHybridR code, it shows that non-thermal particles produce a significant upstream heat flux that is counterbalanced by an enthalpy flux, allowing a kinetic-based closure of the Rankine-Hugoniot conditions. The resulting modified jump conditions predict a higher compression ratio and slower shock speeds, with a flatter non-thermal spectrum at low Mach numbers, in good agreement with simulations and offering a simple, practically implementable closure for fluid models. These findings have broad implications for space weather, CME propagation, and astrophysical shocks, potentially explaining discrepancies between observations (e.g., CME arrival times and radio/X-ray spectra in clusters) and standard fluid predictions, and they highlight the need to include heat-flux effects in predictive models of shock evolution.

Abstract

Collisionless plasma shocks are a common feature of many space and astrophysical systems and are sources of high-energy particles and non-thermal emission, channeling as much as 20\% of the shock's energy into non-thermal particles. The generation and acceleration of these non-thermal particles have been extensively studied, however, how these particles feed back on the shock hydrodynamics has not been fully treated. This work presents the results of self-consistent hybrid particle-in-cell simulations that show the effect of self-generated non-thermal particle populations on the nature of collisionless, quasi-parallel shocks. They contribute to a significant heat flux density upstream of the shock. Non-thermal particles downstream of the shock leak into the upstream region, taking energy away from the shock. This increases the compression ratio, slows the shock down, and flattens the non-thermal population's spectral index for lower Mach number shocks. We incorporate this into a revised theory for the Rankine-Hugoniot jump conditions that include this effect and it shows excellent agreement with simulations. The results have the potential to explain discrepancies between predictions and observations in a wide range of systems, such as inaccuracies of predictions of arrival times of coronal mass ejections and the conflicting radio and x-ray observations of intracluster shocks. These effects will likely need to be included in fluid modeling to accurately predict shock evolution.

Figures (4)

• Figure 1: (a) The reduced ion energy distribution for a $2 d_i$ slice in the $y$ direction as a function of shock normal direction and energy. (b) - (d) Ion distribution functions at the locations shown by the dashed lines in (a), scaled by $|\mathbf{v} - \mathbf{u}|^2$. The distribution functions are plotted in the $v_x$, $v_y$ plane and integrated over the $v_z$ direction. All the distribution functions are averaged in time over $50 \Omega_{ci}^{-1}$ at $0.5 \Omega_{ci}^{-1}$ intervals and are determined in the shock frame.
• Figure 2: (a): predictions for the shock compression ratio as a function of the normalized non-thermal pressure ($\xi$) from different theories as given in the legend. The black dashed lines show the simulation measured non-thermal pressure (vertical) and compression ratio (horizontal), and the theory presented in this work in teal. (b): a scatter plot for the different compression ratio predictions for 4 different shock simulations as a function of measured non-thermal pressure ($\xi$) and Mach number. The red triangle denotes the simulation presented in the other figures. (c): the ion number density plotted as a function of shock normal distance (time averaged as in Figure \ref{['fig:M']} & \ref{['fig:avg_jumps']}). The insert shows the average downstream distribution function (multiplied by $p^4$) as a function of momentum. The predicted non-thermal power law indices are shown by the dashed lines based on standard DSA theory from the corresponding compression ratios.
• Figure 3: Time-averaged Rankine-Hugoniot jump conditions. 1D cuts normal to the shock and centered around the shock front time-averaged as discussed in Figure \ref{['fig:M']}. The black lines in each of the panels correspond to the sum of the fluxes in Eq. \ref{['eq:mass']}, \ref{['eq:momentum']}, and \ref{['eq:energy']} for panel (a), (b), and (c) respectively. The colored lines correspond to the different terms in each of the equations and the brown lines show the non-thermal contribution to the pressure and enthalpy flux density separately. Panel (d) shows the sum (black line) of the heat flux density (orange) and the total enthalpy flux density (purple). The two almost perfectly counterbalance each other in the upstream region, showing that the upstream, non-thermal particles do not contribute to the jump conditions.
• Figure 4: The non-thermal pressure and the total ion number density (divided by 10). 1D cuts are normal to the shock and centered around the shock front and are time-averaged as discussed in Figure \ref{['fig:Pnt']}.