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 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.

💡 Research Summary

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The solar chromosphere is a highly dynamic, partially ionized layer where radiative processes dominate the energy balance. Strong spectral lines, especially the hydrogen Lyman series, are the primary channels for radiative cooling, yet their treatment in three‑dimensional magnetohydrodynamic (MHD) simulations has been limited by computational cost. In the original MURaM extension (Przybylski et al. 2022) non‑Lyman hydrogen lines were handled with a simplified Planck‑function radiation temperature (Sollum 1999) and the Lyman transitions were assumed to be in detailed radiative balance, i.e., the radiative rates were set to zero. This approximation works reasonably well in the lower chromosphere but fails in the upper chromosphere where Lyman‑α photons escape, carrying away a substantial amount of energy.

In this paper the authors present a new module for MURaM that explicitly solves the non‑LTE radiative transfer (RT) for the two strongest Lyman lines, Lyman‑α and Lyman‑β, and couples the resulting radiative rates back into the time‑dependent hydrogen population equations and the energy equation. To keep the computational load manageable, they adopt a single‑frequency approximation originally proposed by Golding et al. (2016). In this approach each line is represented by a single frequency ν₀ and a user‑defined bandwidth Wν; the opacity and source function are averaged over this band, assuming a uniform distribution of photons. The averaged opacity χ̄ is expressed in terms of the lower‑level population, the Einstein B coefficient, and the bandwidth, while the averaged source function S̄ depends on the upper‑level population and the Einstein A coefficient. The radiative transfer equation dI/dτ = S – I is then solved with a 3‑D short‑characteristics solver, yielding the mean intensity J̄ and, consequently, the upward radiative rate Rlu = Blu J̄. The downward rate is simply the spontaneous decay rate Aur. The heating/cooling contribution of each line follows Q = hν₀ ( n_l Rlu – n_u Rul ).

The hydrogen population update retains the previous treatment of advection (partial donor‑cell method) and collisional transitions (using Johnson 1972 cross‑sections). The new radiative rates replace the zero‑rate assumption for Lyman‑α and Lyman‑β, while all other Lyman lines remain under detailed balance because their impact is negligible. The total radiative loss term Q_H from hydrogen lines now includes the self‑consistent contribution from the two solved lines; additional loss terms from Mg II, Ca II, coronal optically thin radiation, and back‑heating are computed as in the earlier version.

A pressure‑based switch limits the non‑LTE RT to regions where the gas pressure p < P_NLTE‑RT (default 10⁵ dyn cm⁻²) and applies the computed rates only where p < P_RT‑USE (default 10⁴ dyn cm⁻²). This ensures a smooth transition to the LTE photosphere and avoids double‑counting of radiative losses already handled by the multigroup photospheric RT.

The authors validate the implementation against the Lightweaver framework, which solves the full frequency‑dependent RT in 1‑D and 3‑D. The single‑frequency model reproduces Lyman‑α and Lyman‑β radiative rate coefficients within 5 % and the associated heating/cooling rates within 10 % across a range of chromospheric conditions. In the lower chromosphere the hydrogen level populations approach detailed radiative balance, while in the upper chromosphere they remain significantly out of balance, matching expectations from observations and previous detailed studies. Convergence tests show that the module remains stable for time steps constrained by the Courant–Friedrichs–Lewy (CFL) condition (Δt ≈ 0.02 s in typical simulations). Because only a single frequency per line is used, the extra computational cost is modest—about a factor of 2–3 increase over the previous version, far less than a full multi‑frequency treatment would require.

The paper demonstrates that incorporating a physically realistic treatment of the Lyman series dramatically improves the fidelity of chromospheric simulations. Electron densities, H I fractions, and temperature structures in the upper chromosphere become more consistent with spectroscopic observations. Moreover, the modular design of the single‑frequency RT allows straightforward extension to other important species (He I, Ca II, Mg II), paving the way for comprehensive non‑LTE chromospheric modeling in high‑resolution 3‑D MHD simulations.

In summary, the authors have successfully integrated a self‑consistent, non‑LTE radiative transfer module for the Lyman‑α and Lyman‑β lines into the MURaM code using a computationally efficient single‑frequency approximation. The approach yields accurate radiative rates, improves hydrogen population dynamics, and maintains numerical stability under realistic CFL‑limited time steps, thereby enabling more realistic simulations of chromospheric dynamics and energetics.