Kinetic collisionless model of the solar transition region and corona with spatially intermittent heating

TL;DR

The paper addresses how the solar transition region and corona can arise without direct coronal heating by developing a 3D kinetic model in which the coronal plasma is collisionless and embedded in a uniform magnetic field. Localized heating events distributed over a finite chromospheric surface are incorporated via a surface coarse-graining that yields non-thermal boundary conditions for the Vlasov equation, leading to stationary distribution functions with suprathermal tails and temperature inversion through velocity filtration. A key result is that the temperature increases with height and the density decreases due to the combined effects of gravity and the suprathermal component; this inversion is robust to the precise form of the boundary heating distribution $oldsymbol{b3(T)}$ and depends mainly on the heating-area fraction $A$. The findings provide a spatial-intermittency mechanism for the formation of the transition region and corona, offer analytical expressions for $n(z)$ and $T(z)$, and point to future extensions including magnetic-field variations, collisions, and applications to other stars, with a transition-region width of order $z oughsim 3 imes10^3$ km in typical cases.

Abstract

We develop a three-dimensional kinetic model of the solar transition region and corona in which the plasma above the chromosphere is collisionless and embedded in a uniform magnetic field. Heating occurs intermittently at discrete locations on the chromospheric surface, modeled through a surface coarse-graining procedure that produces non-thermal boundary conditions for the Vlasov equation. The resulting stationary distribution functions generate suprathermal particle populations and naturally lead to a temperature inversion via gravitational filtering, without any local coronal heating. The model reproduces realistic temperature and density profiles with a thin transition region and a hot corona, consistent with solar observations. These results demonstrate that the spatial intermittency of heating at the chromospheric interface is sufficient to account for the formation of the transition region and the high-temperature corona.

Figures (4)

• Figure 1: Schematic representation of the solar plasma model. The surface $S$, located at the base of the transition region, acts as the interface between the fully collisional chromosphere (serving as a thermal reservoir) and the collisionless coronal plasma. Localized heating events, each occupying an area $s_H$, are shown in black. The coronal plasma above is embedded in a uniform magnetic field $\textbf{B}$ (green), while particles are subject to a net external force $\textbf{g}(m_p + m_e)/2$ (black), which combines gravitational field and the Pannekoek–Rosseland electric field. The Cartesian reference frame $(x, y, z)$ used throughout the paper is indicated in the top right.
• Figure 2: The three probability distribution functions $\gamma(T)$ used in next figures. $\gamma_1(T)$ is defined by Eq. \ref{['exponentialincrements']} (continuous line), $\gamma_2(T)$ by Eq. \ref{['halfgaussianincrements']} (dashed line), and $\gamma_3(T)$ by Eq. \ref{['twosidedgaussian']} (dotted line). The parameters are $T_0 = 10^4$ K, and $\Delta T = 90\, T_0$. For the distribution $\gamma_3(T)$, we choose $T_h = \Delta T = 90\, T_0$, with $T_R = T_h$ and $T_L = 0.1\, T_h$ to satisfy the observational constraints.
• Figure 3: Right panel: number density profiles $\mathrm{cm}^{-3}$ as function of the height expressed in km computed using the distribution of heating events $\gamma_1(T)$ given by Eq. \ref{['exponentialincrements']} (solid), $\gamma_2(T)$ given by Eq. \ref{['halfgaussianincrements']} (dashed) and $\gamma_3(T)$ given by Eq. \ref{['twosidedgaussian']} (dotted). Blue lines correspond to $A = 1$, red lines to $A = 0.1$, green lines to $A = 0.01$ and purple lines to $A=0.001$. Right panel: temperature profiles $T[K]$ as functions of the height expressed in km. The profiles are computed for the same distribution of temperature increments and values of $A$ of the left column. Moreover the same color coding and line style has been used.
• Figure 4: Decimal logarithm of the velocity distribution functions (VDFs), for species $\alpha$, plotted as a function of the signed kinetic energy normalized by $v_{T_0,\alpha}^2$. The VDFs are scaled by the corresponding number densities and normalized by $v_{T_0,\alpha}^3$, where $v_{T_0,\alpha}$ is defined by Eq. \ref{['v_T_0']}. Each panel compute the VDFs for different values of $A$ as shown in the subplot title. In each panel the VDFs are computed using the three distribution of temperature increments $\gamma_1(T)$ defined in Eq.\ref{['exponentialincrements']} (solid), $\gamma_2(T)$ defined in Eq.\ref{['halfgaussianincrements']} (dashed) and $\gamma_3(T)$ defined in Eq.\ref{['twosidedgaussian']} (dotted). Finally, in each panel, the VDFs are shown at three different heights (see their positions in Fig. \ref{['fig:temperatureinversion']}): $z_1 = 2 \times 10^3$ km (base location, blue), $z_2 = 3.5 \times 10^3$ km (transition region, red), and $z_3 = 14 \times 10^3$ km (corona, green).