Non-thermal particle acceleration in multi-species kinetic plasmas: universal power-law distribution functions and temperature inversion in the solar corona

TL;DR

This work develops a self-consistent quasilinear theory for electromagnetically driven, multi-species kinetic plasmas, deriving a Fokker-Planck transport equation that couples drive heating with Balescu-Lenard diffusion and drag from internal turbulence and collisions. A key result is that Debye screening of large-scale, super-Debye but sub-Larmor fields yields a universal $v^{-5}$ tail in both electron and ion distributions, corresponding to an $E^{-2}$ energy spectrum, robust against weak collisions. When applied to the solar corona, the theory explains the observed sharp chromosphere-to-corona temperature rise via velocity filtration, with κ-distributions in the range $1.5\lesssim\kappa\lesssim2$ reproducing the data, and a double-$\kappa$ model improving fits for heavy ions. These findings link microphysical diffusion processes to macroscopic temperature structure and suggest broad applicability of the mechanism to other weakly collisional plasmas, including winds, shocks, and accretion environments. Future work will extend the framework to magnetized, relativistic regimes and explore energy partition between species.

Abstract

The origin of non-thermal power-law distribution functions ubiquitously observed in astrophysical/space (e.g., the solar wind) and laboratory kinetic plasmas, is not well understood. Another puzzling phenomenon is temperature inversion in the solar corona. These two issues are deeply connected. We develop a self-consistent quasilinear theory (QLT) for electromagnetically driven kinetic plasmas, deriving a Fokker-Planck equation for the simultaneous relaxation of multiple species, with (i) a drive diffusion coefficient for the heating of dressed particles directly by the drive and indirectly by waves, and (ii) Balescu-Lenard diffusion and drag coefficients for internal turbulence and Coulomb collisions. Both electron and ion distributions relax towards a universal attractor with a $v^{-5}$ $(E^{-2})$ tail, akin to a $κ= 1.5$ distribution, under a super-Debye (but sub-Larmor) drive with a steep power-spectrum. This is an outcome of Debye screening: large-scale fields accelerate the unscreened, fast particles but not the screened, slow ones. The universality may be broken by shallow power-spectra and incomplete relaxation. Collisions cannot decelerate suprathermal particles, rendering a high $v$ tail immune to Maxwellianization. Such a tail may be generated in the solar corona by chromospheric convection despite collisional losses. The suprathermal particles escape sun's gravity (velocity filtration), inverting the temperature profile and raising it to $10^6$ K. A proper analysis of velocity filtration with a $κ\approx 1.5-2$ distribution inspired by QLT provides a reasonable fit to the spectroscopic data of heavy ions and explains the abrupt temperature rise, a consequence of the divergence of pressure in the $κ\to 1.5$ limit.

Figures (7)

• Figure 1: Electron diffusion coefficient due to direct dressed particle heating by the drive (solid lines) and indirect heating by the waves (dashed lines), in units of $D^{(0)}_e = \left(32\pi^5 e^2 k^2 \calE_0/m_e^2 V\right) \left[(3-\alpha)/\left(1 - {\left(k_{\rm min}/k_\rmc\right)}^{3-\alpha}\right)\right]$, vs $v$ for different values of $\alpha$, with the electric field power spectrum of the drive given by $\calE(k)\sim k^{-\alpha}$ (see equation [\ref{['Schechter_func']}]). Ion diffusion coefficient scales similarly with $v$. The plasma is assumed to be marginally stable. We adopt $T_e = T_i$, $k_{\rm min}\lambda_\rmD = 10^{-3}$, and $\omega_{\rmP e}t_\rmc = 10$. Note that the dressed particle coefficient $D_p^{(\rms)}(v)$ typically dominates over the wave coefficient $D_w^{(\rms)}(v)$. For $\alpha \geq 5$, $D_p^{(\rms)}(v)$ scales as $v^4$ at $\sigma_e < v < 1/k_{\rm min}t_\rmc$ and as $v^{\alpha - 2}$ at $1/k_{\rm min} t_\rmc < v < \omega_{\rmP e}/k_{\rm min}$, while, for $\alpha < 5$, it scales as $v^{\alpha - 1}$ at $\sigma_e < v < \omega_{\rmP e}/k_{\rm min}$. Both $D_p^{(\rms)}(v)$ and $D_w^{(\rms)}(v)$ fall off as $v^{-3}$ beyond $\omega_{\rmP e}/k_{\rm min}$.
• Figure 2: Diffusion and drag coefficients vs $v$ for electrons, in units of $D^{(0)}_e = 32\pi^5 e^2 k^2 \calE_0/m_e^2 V$, assuming $\calE(k') = \calE_0\,\delta(k'-k)$, $k\lambda_{\rmD e} = 10^{-2}$ and $\omega_{\rmP e}t_\rmc = 1$. The ion coefficients scale similarly with $v$. Solid, dashed and dot-dashed lines indicate the drive diffusion, BL diffusion and BL drag coefficients respectively. Blue (red) lines represent $T_e = T_i$$(T_e = 10^2 T_i)$. Note that the drive diffusion coefficient $D^{(e)}(v)$ scales as $v^4$ in the range $\sigma_e = \sqrt{k_\rmB T_e/m_e}<v<\omega_{\rmP e}/k$ for $T_e = T_i$ (also at $T_i > T_e$), and similarly at $\sigma_i=\sqrt{k_\rmB T_i/m_i}<v<c_\rms = \sqrt{k_\rmB T_e/m_i}$ for $T_i < T_e$. The BL diffusion (drag) coefficient is constant (scales as $v$) at small $v$ and scales as $v^{-3}$$(v^{-2})$ at large $v$. Note that the drive coefficient always wins over the BL ones at large $v$ for a super-Debye scale drive.
• Figure 3: Development of the non-thermal power-law tail in an initially Maxwellian ion distribution due to large-scale EM turbulence, obtained by solving equation (\ref{['QL_eq']}). We adopt $D^{(i)}(v = 0) = 10^{-2} \calD^{(i)}_2(v = 0)$, $\sigma_i = 0.1\sigma_e$, $\alpha = 8$, $k_{\rm min} \lambda_{\rmD i} = 10 k_{\rm min} \lambda_{\rmD e} = 10^{-3}$, and $\omega_{\rmP i} t_\rmc = 0.1 \omega_{\rmP e} t_\rmc = 1$. Note the Maxwellianization of the bulk around $\sigma_i$ and the emergence of the $v^{-5}$ tail beyond, followed by a $v^{1-\alpha} \sim v^{-7}$ tail and a Maxwellian fall-off.
• Figure 4: Same as Fig. \ref{['fig:fi_vs_t']} but for electrons. We adopt $D^{(e)}(v = 0) = \calD^{(e)}_2(v = 0)$, $\alpha = 8$, $k_{\rm min} \lambda_{\rmD e} = 10^{-3}$, and $\omega_{\rmP e} t_\rmc = 10$. Just as in the ions, a $v^{-5}$ tail develops, followed by a $v^{1-\alpha} \sim v^{-7}$ tail and a Maxwellian fall-off.
• Figure 5: Maxwellianization of the electron DF, initialized as a $\kappa = 1.5$ distribution ($v^{-5}$), after the EM drive has been switched off and the plasma relaxes due to turbulent and collisional relaxation through the BL equation (equation [\ref{['QL_eq']}] without the drive). Note the continued presence of the $v^{-5}$ tail of runaway electrons.