Effect of flow-aligned external magnetic fields on mushroom instability

TL;DR

This study addresses how a flow-aligned external magnetic field influences mushroom instability (MI) in magnetized relativistic shear flows. Using a cold, collisionless two-fluid framework, the authors derive a generalized MI dispersion relation and show, via numerical solutions and 2D PIC simulations, that an external field consistently suppresses MI growth, though MI is more robust than electron-scale Kelvin-Helmholtz instabilities (ESKHI) and lacks a finite stabilization threshold in the cold limit. They also extend the analysis to coupled instabilities with arbitrary in-plane wavevectors, revealing field-induced modifications but that the MI peak persists at $k_y=0$ in relativistic regimes. PIC simulations corroborate the theory: magnetized runs exhibit slower MI growth, quasi-steady saturation, and a diffusion-induced DC magnetic field in finite-temperature cases that can dominate early dynamics. Overall, the work suggests MI remains a principal mechanism for magnetic field amplification and jet spine dynamics even in strongly magnetized environments, with implications for jet collimation and magnetic structure formation.

Abstract

Mushroom instability (MI) is a shear instability considered responsible for generating and amplifying magnetic fields in relativistic jets. While astrophysical jets are usually magnetized, how MI acts in magnetized jets remains poorly understood. In this paper, we investigate the effect of a flow-aligned external magnetic field on MI, with both theoretical analyses and particle-in-cell (PIC) simulations. In the limit of a cold and collisionless plasma, we derive a generalized dispersion relation for linear growth rates of the magnetized MIs. Numerical solutions of the dispersion relation reveal that the external magnetic field always suppresses the growth of MI, though MIs are much more robust against the external magnetic field than electron-scale Kelvin-Helmholtz instabilities (ESKHIs). Analyses are also extended to instabilities with an arbitrary wavevector in the shear interface plane, where coupling effect is observed for sub-relativistic scenarios. Two-dimensional PIC simulations of single-mode MIs reach a good agreement with our analytical predictions, and we observe formation of a quasi-steady saturation structure in magnetized runs. In simulations with finite temperatures, we observe the competition and cooperation between MIs and a diffusion-induced DC magnetic field.

Figures (11)

• Figure 1: The dispersion relation of the MI growth rate $\sigma(k_z)$ with $v_0=V_0 \text{sgn}(x)$ and $V_0=0.2c$ under uniform external magnetic field with different magnitudes $B_0$ corresponding to $\omega_c/\omega_p=eB_0/m_e\gamma_0\omega_p=$0.5 (yellow), 1 (green), 2 (red), 3 (purple), 5(brown), and 10 (cyan). Each curve is interpolated by numerically solving the eigenequation at intervals of $\Delta k_zV_0=\omega_p/20$. The theoretical dispersion relation in absence of external magnetic field, Eq. \ref{['eq:alves']} (blue and dashed), is also shown for comparison.
• Figure 2: The same plot as Fig. \ref{['fig:dis02v']}, but with $V_0=0.5c$. Results extracted from single-mode PIC simulations are also marked for comparison, where the horizontal coordinate of the marks corresponds to the wavenumber of the pre-imposed perturbation, and the vertical coordinate corresponds to the growth rates extracted from the simulations. The details of the simulations will be discussed in Sec. \ref{['sec:sim']}.
• Figure 3: The same plot as Fig. \ref{['fig:dis02v']}, but with $V_0=0.8c$.
• Figure 4: Normalized maximum growth rates $\sigma_\text{max}/\omega_{p0}$ of MIs (solid lines) and ESKHIs (dashed lines) as functions of the magnitude of the normalized external magnetic field $\omega_{c0}/\omega_{p0}=eB_0/m_e\omega_{p0}$, with $V_0/c=$0.2 (blue), 0.5 (yellow) and 0.8 (green). For consistent comparison, here $\sigma_\text{max}$ and $B_0$ are normalized with the non-relativistic frequencies $\omega_{c0}=eB_0/m_e$ and $\omega_{p0}=\sqrt{n_{e0} e^2/m_e\epsilon_0}$, which only depend on $B_0$ and $n_{e0}$ respectively and remain invariant with different $V_0$ or $\gamma_0$. They are related to our previously defined relativistic frequencies by $\omega_c=\omega_{c0}/\gamma_0$ and $\omega_p=\omega_{p0}/\gamma_0^{3/2}$.
• Figure 5: The dispersion relation of the coupled shear instability linear growth rate $\sigma(k_y,k_z)$, in a symmetrical shear flow with $V_0=0.2c$, under two different magnitudes of external magnetic field $\omega_c/\omega_p=$ 0 (blue) and 0.5 (red).