BO-PBK: A comprehensive solver for dispersion relations of obliquely propagating waves in magnetized multi-species plasma with anisotropic loss-cone drift product-bi-kappa distribution

TL;DR

The paper tackles the challenge of analyzing waves and instabilities in magnetized multi-species plasmas that follow non-Maxwellian, anisotropic velocity distributions, such as PBK, KM, and BM with loss-cone. It introduces BO-PBK, an eigenvalue-based solver that reformulates the linearized Vlasov-Maxwell equations into a compact dispersion-function–driven eigenproblem, enabling one-shot computation of multiple wave branches without iterative searches. A unified, rational-form susceptibility is derived using the PBK framework and its Maxwellian limits, and the full system couples PBK, BM, and Maxwell equations into a sparse closed linear system with adaptive switching to BM to manage matrix size. Benchmarks against established kinetic theories demonstrate high accuracy and substantial efficiency gains, making BO-PBK a practical tool for space- and laboratory-plasma wave analysis and stability studies, with open-source availability at the provided repository.

Abstract

We present BO-PBK (BO-Product-Bi-Kappa), a new solver for kinetic dispersion relations of obliquely propagating waves in magnetized plasmas with complex velocity distributions. It reformulates the linearized Vlasov-Maxwell system into a compact eigenvalue problem, enabling direct computation of multiple wave branches and unstable modes without iterative initial-value searches. Key innovations include a unified framework supporting product-bi-kappa, kappa-Maxwellian, bi-Maxwellian, and hybrid distributions with multi-component and loss-cone features; a concise rational-form eigenvalue formulation; and a 2--3 times reduction in matrix dimensions compared to the BO-KM solver, with improved efficiency at larger kappa indices. Benchmark tests confirm accurate reproduction of standard kinetic results and efficient resolution of waves and instabilities. BO-PBK thus provides a computationally efficient tool for wave and stability analysis in space and laboratory plasmas.

Figures (12)

• Figure 1: Contours of the normalized electron velocity distribution functions in the ($v_{\parallel}$, $v_{\perp}$) plane with $T_{\parallel e} = T_{\perp e} = 1\times 10^2$ eV. Panels (a)-(c) show the PBK, KM, and BM distributions without a loss-cone, while panels (d)-(f) show the corresponding distributions with a loss-cone $\sigma = 0.5$.
• Figure 2: Normalized electron velocity distribution functions for Kappa and Maxwellian models at $T_{\parallel e} = T_{\perp e} = 1\times 10^2$ eV. Left: parallel distributions $f_{\parallel}^\mathrm{K}(v_{\parallel})$ and $f_{\parallel}^\mathrm{M}(v_{\parallel})$; right: perpendicular distributions $f_{\perp}^\mathrm{K}(v_{\perp})$ and $f_{\perp}^\mathrm{M}(v_{\perp})$ with a loss-cone at $\sigma=0.5$.
• Figure 3: $\omega_r/|\omega_{ce}|$ vs. $\rho_{ce}$ (top) and $\omega_i/|\omega_{ce}|$ vs. $\rho_{ce}$ (bottom) for $\kappa_{\parallel e}=1$, $\kappa_{\perp e}=200$, $\theta=30^\circ$, $\tilde{\theta}_{\perp e}/c=0.1$, $\omega_{pe}/|\omega_{ce}|=0.5$, $T_{\perp e}/T_{\parallel e}=1$, and $B_0=1.0\times 10^{-6}$ T, comparison with data from Fig. 1 of Cattaert et al. Cattaert2007 (black dashed lines).
• Figure 4: Comparison of real frequency $\omega_r$ (top) and growth rate $\omega_i$ (bottom) of parallel electron whistler-cyclotron modes for $\kappa_{\parallel,\perp(e,p)}=2,6,\infty$ in weakly magnetized plasmas ($|\omega_{ce}|/\omega_{pe} = 0.01$), with $v_{T\parallel e} = 0.02c$ and $T_{\perp(e,p)}/T_{\parallel(e,p)} = 4$. Data from Fig. 2 of Lazer et al. Lazar_Poedts2010, are shown as black dashed lines.
• Figure 5: Comparison of whistler wave real frequency $\omega_r$ vs. $k$ (top) and growth rate $\omega_i$ vs. $k$ (bottom) with varying temperature anisotropy. Parameters: $\kappa_{\parallel e} = 1$, $\kappa_{\perp e} = 200$, $\theta = 30^{\circ}$, $\theta_{\perp e}/c = 0.1$, $|\omega_{ce}|/\omega_{pe} = 0.5$, and $B_0 = 1.0 \times 10^{-6}$ T. Comparison with data from Fig. 9 of Cattaert et al. Cattaert2007 shown as dashed lines.