A single frequency approach to nonequilibrium modeling of the chromosphere

TL;DR

The paper addresses the need to model NLTE radiative transfer in the solar chromosphere, focusing on Lyman-series lines that dominate radiative energy exchange. It introduces an updated MURaM version that solves NLTE radiative transfer for Lyman-α and Lyman-β with a single-frequency approximation, coupling the RT solution to hydrogen level populations and temperature evolution self-consistently. The results show good agreement with the Lightweaver reference and reveal a hotter upper chromosphere and a more extended partially ionized region when NLTE Lyman RT is included, with populations closer to detailed radiative balance away from wavefronts. The approach maintains computational efficiency suitable for high-resolution 3D simulations and provides a path to extending NLTE RT to additional species and lines for chromospheric energy balance.

Abstract

The solar chromosphere is a region where radiation plays a critical role in energy transfer and interacts strongly with the plasma. In this layer, strong spectral lines, such as the Lyman lines, contribute significantly to radiative energy exchange. Due to the long ionization/relaxation timescale, departures from LTE become significant in the chromosphere. Accurately modeling this layer therefore requires one to solve the non-LTE radiative transfer for the Lyman transitions. We present an updated version of the MURaM code to enable more accurate simulations of chromospheric hydrogen level populations and temperature evolution. In the previous extension, a non-LTE equation of state, collisional transitions of hydrogen, and radiative transitions of non-Lyman lines were already implemented in the code. Building on this, we have now incorporated radiative transfer for the Lyman lines to compute radiative rate coefficients and the associated radiative losses. These were used to solve the population and temperature evolution equations, rendering the system self-consistent. To reduce computational cost, a single-frequency approximation was applied to each line in the numerical solution of the radiative transfer problem. The extended model shows good agreement with reference solutions from the Lightweaver framework, accurately capturing the radiative processes associated with Lyman lines in the chromosphere. The extension brings the simulated hydrogen level populations in the deep chromosphere closer to detailed radiative balance, while those in the upper chromosphere remain significantly out of balance, consistent with the expected conditions in the real solar atmosphere. The extension enables the MURaM code to accurately capture chromospheric dynamics.

Figures (8)

• Figure 1: Schematic illustration of the mesh concerning RT. The left panel shows the computational meshes employed for solving the governing equations in the MURaM code, while the right panel illustrates the relationship between the radiation ray and the mesh structure. The MHD mesh is employed for solving the MHD equations, whereas the RT mesh is used for solving the RT equation. Both meshes are uniform and Cartesian, with the RT grid points positioned at the centers of the cells formed by adjacent MHD grid points. Information exchange between the two meshes is accomplished through bi-linear or tri-linear interpolation. Rays are involved in solving the RT equation using the short-characteristics method, in which the ray intensity at the grid points ($\mathrm{o}$) is the quantity being solved for in MURaM. The solution involves the source functions at the upwind ($\mathrm{u}$), downwind ($\mathrm{d}$), and grid points, along with the ray intensity at the upwind point. The rays typically do not intersect with the grid vertices. Linear or bi-linear interpolation was employed to compute the source function, opacity, and ray intensity at the upwind and downwind points.
• Figure 2: Temperature and $n_2$ population distributions at $t = 60\,\mathrm{s}$ for the four simulation cases. The average time steps of the CFL = 0.6 and CFL = 0.1 cases are approximately $10^{-2}\,\mathrm{s}$ and $2 \times 10^{-3}\,\mathrm{s}$, respectively. Panels (a) and (b) display the temperature and $n_2$ distributions for the case with CFL = 0.6. Panels (c) and (d) show the corresponding regions (outlined in panels (a) and (b)) for all four cases, respectively. Panels (e) and (f) present 1D profiles of temperature and $n_2$, respectively, extracted along the dashed lines in panels (a) and (b), for all four cases. In panel (f), an additional $n_2$ profile computed from the balance solution for the $\Delta t = 10^{-5}\,\mathrm{s}$ case is also shown for reference.
• Figure 3: Comparison of the upward radiative rate coefficients obtained by MURaM (using the single-frequency approach) and reference (using the multifrequency approach by Lightweaver). Panels (a) and (b) show the atmospheric density and temperature distributions, respectively. Panels (c) and (d) display the Lyman-$\alpha$ and Lyman-$\beta$ rate coefficients in the cool region ($T<=0.2\ \mathrm{MK}$) computed by MURaM, while panels (e) and (f) present the corresponding coefficients from the reference. Panels (g) and (h) display the results of the point-to-point amplitude comparison, with the horizontal axis representing the values from MURaM and the vertical axis corresponding to the amplitudes from the reference. The points are colored according to their corresponding color in the MURaM result panels (c, d), so the area where the points are located can be inferred from the color. Since the highly ionized corona is not the region of interest for NLTE RT, the rate coefficients at high temperatures ($T>0.2\ \mathrm{MK}$) were manually set to zero to emphasize the comparison results in the lower atmosphere. The time corresponding to the simulated data is $t = 1\ \mathrm{s}$.
• Figure 4: Comparison of the upward radiative rate coefficients obtained by MURaM and Lightweaver (reference). Panels (a) and (b) show the heating effects contributed by the Lyman-$\alpha$ and Lyman-$\beta$ lines from MURaM, while panels (c) and (d) present the corresponding heating effects in the reference. Panels (e) and (f) give the point-to-point amplitude comparison, where the horizontal axis corresponds to the values obtained from MURaM and the vertical axis to the amplitudes from the reference. The points are colored based on their sign in both the MURaM and reference results, with red and pink indicating heating, and blue and cyan indicating cooling. The time corresponding to the simulated data is $t = 1\ \mathrm{s}$.
• Figure 5: Comparison of the temperature and population distributions obtained by two versions of the MURaM code. The left column shows results from the previous version, while the right column presents results from the updated version. The first row displays the temperature distribution, while the second, third, fourth, and fifth rows show the population fractions of level 1, level 2, level 3, and protons, respectively, where $n_\mathrm{p}$ is the proton number density and $n_\mathrm{H}$ is the total hydrogen number density. To compare the extended height of the incompletely ionized region, a reference line is provided in panels (c), (d), (i) and (j). The previous version assumes $R_{lu}=R_{ul}=0$ for the Lyman series and computes radiative cooling using an empirical recipe, whereas the updated version calculates the Lyman-$\alpha$ and $\beta$ lines rate coefficients and the associated radiative heating and cooling through solving the RT equation. The time corresponding to the simulated data is $t = 20\, \mathrm{s}$