A single frequency approach to nonequilibrium modeling of the chromosphere

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 local thermodynamic equilibrium (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. Convergence tests show that the module can accurately capture the evolution of temperature and hydrogen level populations with simulation time steps constrained by the Courant–Friedrichs–Lewy (CFL) condition. The extension enables the MURaM code to accurately capture chromospheric dynamics. Its robust performance under large simulation time steps renders it particularly well suited for high-resolution, three-dimensional simulations.