Thermal wakefield structure in plasma acceleration processes: insights from fluid models and PIC simulations

TL;DR

This work analyzes how non-negligible plasma temperature affects plasma wakefield acceleration by comparing two thermal fluid closures, Local Equilibrium Closure ($ ext{LEC}$) and Warm Plasma Closure (WARMC), against fully kinetic PIC data. Using spatially resolved simulations, it characterizes the first electron depletion bubble through its longitudinal and transverse sizes $(\,\ell_{\parallel}, \ell_{\perp})$ and examines the accelerating and focusing wakefields inside the bubble; WARMC consistently outperforms LEC as $T_i$ increases, up to a regime where nonlinearity and temperature start to degrade accuracy. The study quantifies WARMC’s validity via a temperature threshold $T_{ ext{max}}$ that decreases with the wake’s nonlinearity parameter $ ilde{Q}$, showing that longitudinal observables are more sensitive to thermal effects than transverse ones. The results provide practical guidance for fluid PWFA simulations, and point to hybrid or higher-moment closures as promising avenues to extend accuracy in warm, nonlinear regimes.

Abstract

We focus on the process of plasma acceleration in the presence of non-negligible thermal effects, wherein a driver of relativistic electrons perturbs a warm neutral plasma and generates a wakefield structure. We study the acceleration process via numerical simulations based on fluid models with different thermal closure assumptions, and also provide systematic comparisons against ground-truth data coming from particle-in-cell (PIC) simulations. The focus of the analysis is on the first electron depletion bubble after the driver, where we provide a detailed characterization of its size and the electromagnetic fields developed inside. Our results are instrumental in determining the correct thermal closure assumption to be used in fluid models for the numerical simulations of plasma acceleration processes, as well as elucidating the corresponding limits of applicability.

Figures (7)

• Figure 1: Snapshots of normalized electron plasma density $n/n_i$ for LEC (panel (a)), WARMC (panel (b)) and PIC (panel (c)). Each panel shows a comparison between the cold case ($k_B T_i = 0~\rm{keV}$, upper half) and a warm case ($k_B T_i = 0.5~\rm{keV}$, lower half). Corresponding values of $\mu_i = k_B T_i/m_e c^2$ are also reported. In panel (a), a sketch is provided to illustrate the longitudinal ($\ell_{\parallel}$) and the transverse ($\ell_{\bot}$) size of the bubble (see text for details). Simulations are performed with $\tilde{Q}= 1.0$. Spatial coordinates are made dimensionless w.r.t. $k_p^{-1}$.
• Figure 2: We analyze $\ell_{\parallel}$ and $\ell_{\bot}$ at changing $k_B T_i$ (or equivalently $\mu_i = k_B T_i/m_e c^2$) for LEC (blue points/lines) and WARMC (red points/lines) at fixed $\tilde{Q}=10^{-4}$. Results of numerical simulations (points) are compared with corresponding theoretical predictions in the linear regime (lines, see text for details). Both $\ell_{\parallel}$ and $\ell_{\bot}$ are normalized w.r.t. the corresponding value in the cold limit ($\ell^{\rm{cold}}_{\parallel}$, $\ell^{\rm{cold}}_{\bot}$).
• Figure 3: Panels (a) and (b): we analyze $\ell_{\parallel}$ and $\ell_{\bot}$ at changing $k_B T_i$ (or equivalently $\mu_i = k_B T_i/m_e c^2$) for LEC (blue points), WARMC (red points) and PIC (orange triangles) data at fixed $\tilde{Q}=1.5$. Both $\ell_{\parallel}$ and $\ell_{\bot}$ are normalized w.r.t. the corresponding value in the cold limit ($\ell^{\rm{cold}}_{\parallel}$, $\ell^{\rm{cold}}_{\bot}$). Error-bars in PIC data are obtained by averaging 9 independent simulations. Panel (c): we analyze $k_B T_{\rm{max}}$ (or equivalently $\mu_{\rm{max}}$) at changing $\tilde{Q}$. $T_{\rm{max}}$ is the temperature at which the relative discrepancy in $\ell_\parallel$ between WARMC and PIC reaches a threshold value (see text for details).
• Figure 4: We analyze $\ell_{\parallel}$ and $\ell_{\bot}$ at changing $\tilde{Q}$ for WARMC (circles) ad PIC (triangles) data for different $k_BT_i$: $k_BT_i=0~\rm{keV}$ (blue), $k_BT_i=0.5~\rm{keV}$ (red), $k_BT_i=3.0~\rm{keV}$ (green). Corresponding values of $\mu_i = k_B T_i/m_e c^2$ are indicated. Both $\ell_{\parallel}$ and $\ell_{\bot}$ are made dimensionless w.r.t. $k_p^{-1}$. Panel (a): log-log plot of $k_p \ell_\bot$ as a function of $\tilde{Q}$. We also report the scaling $k_p \ell_\bot \sim \tilde{Q}^{1/2}$ found in the cold limit (dashed black line, see text for details). Panel (b): lin-log plot of $k_p \ell_\parallel$ as a function of $\tilde{Q}$.
• Figure 5: Comparison between WARMC (top-half panels) and PIC (bottom-half) results for the accelerating field $E_z$ (top row, panels (a)-(c)) and the focusing field $E_f=E_r+cB_\phi$ (bottom row, panels (d)-(f)) for different $k_B T_i$: $k_BT_i = 0~\rm{keV}$ (first column, panels (a) and (d)), $k_BT_i = 0.5~\rm{keV}$ (second column, panels (b) and (e)), $k_BT_i = 3.0~\rm{keV}$ (third column, panels (c) and (f)). Corresponding values of $\mu_i = k_B T_i/m_e c^2$ are also indicated. Simulations are performed with $\tilde{Q}=1.0$. Fields and spatial coordinates are made dimensionless w.r.t. $E_0 = m_e c \omega_p / e$ and $k_p^{-1}$ respectively.