Investigating Nonlinear Landau Damping in Hybrid Simulations

TL;DR

The paper investigates nonlinear Landau damping of self-generated turbulence from cosmic-ray streaming in a β≳1 plasma using hybrid-PIC simulations, demonstrating that the damping rate at a given scale depends on the magnetic power on larger scales via $Γ_{NLLD} \propto k \int_0^k \delta B(k')^2 \,dk'$ and that an inverse cascade transfers energy to nonresonant scales with $Γ_g \propto k \int_k^\infty \delta B(k')^2 \,dk'$. It shows that in a periodic box an inverse cascade prevents a steady state, while open boundaries allow energy to escape and a quasi-steady state to emerge (up to the limit of large-scale growth). Extending to multiple CR populations, the results remain consistent with linear two-wave interaction theory, with damping tracking the total large-scale power and spectral peaks shifting as energy cascades to larger scales. The findings have important implications for Galactic CR transport, indicating that large-scale Alfvénic turbulence can suppress self-confinement and necessitate revised damping prescriptions in transport models.

Abstract

Phenomenological studies of cosmic-ray self-confinement often hinge on the linear theory for the growth rate of the streaming instability and for the damping rate of the generated magnetic modes. Largely different expressions exist, especially for the rate of nonlinear Landau damping, which is often assumed to be the most important damping mechanism in warm ionized plasmas. Using hybrid-PIC simulations in the resonant streaming instability regime, we present a comprehensive assessment of nonlinear Landau damping and show that the damping rate at a given scale depends on the power in magnetic fields on larger scales. Furthermore, we find that an inverse cascade develops, which produces magnetic fields on scales larger than the resonant ones. Here we extend previous results obtained for a mono-energetic distribution of non-thermal particles to the case of broader CR distributions, as a first step towards developing phenomenological models. Pre-existing turbulence of Alfvénic nature at large scales severely affects the damping of waves produced by low-energy CRs; depending on its amplitude, such a turbulence may inhibit the growth of streaming instability so that CRs are either self-confined at all energies or not at all.

Figures (5)

• Figure 1: Top panel: Time evolution of the total perpendicular magnetic field (blue) and magnetic field on all scales smaller (orange) and larger (green) than the Larmor radius of CRs. Time is multiplied by the predicted growth rate of the instability. Bottom panel: Box-averaged CR drift velocity as a function of time.
• Figure 2: Time evolution (color coded) of the power in left-handed modes for case $\mathcal{B}$. The vertical line indicates the scale $k_0^{-1}$ for which the power contained on smaller scales is equal to the power in larger scales. In the yellow-shaded region the growth due to the inverse cascade dominates over damping.
• Figure 3: CR distribution function in $x$ momentum at the beginning and end of the simulation of case $\mathcal{B}$.
• Figure 4: Top panel: Evolution of the transverse magnetic fields as in Fig. \ref{['fig:caseII']}, but for case $\mathcal{C}$. Note that in this case the growth rate and resonant wavenumber are a factor $3$ smaller than in the other cases due to the different CR population. Bottom panel: Box averaged CR drift velocity as a function of time normalized by its initial value $v_{D,0}=12\,v_A$.
• Figure 5: Same as Fig. \ref{['fig:caseII']}, but for case $\mathcal{D}$. Since the drift speed is shown for two CR populations, the velocities are normalized to their individual initial drift speed, i.e., $4\,v_A$ and $12\,v_A$ respectively.