Prospects and challenges of numerical modeling of the Sun at millimeter wavelengths

The Atacama Large Millimeter/submillimeter Array (ALMA) offers new diagnostic possibilities that complement other commonly used diagnostics for the study of the Sun. In particular, ALMA's ability to serve as an essentially linear thermometer of the chromospheric gas at unprecedented spatial resolution at millimeter wavelengths and future polarization measurements has great diagnostic potential. Solar ALMA observations are therefore expected to contribute significantly to answering long-standing questions about the structure, dynamics, and energy balance of the outer layers of the solar atmosphere. In this regard, current and future ALMA data are also important for constraining and further developing numerical models of the solar atmosphere, which in turn are often vital for the interpretation of observations. The latter is particularly important given the Sun's highly intermittent and dynamic nature that involves a plethora of processes occurring over extended ranges in spatial and temporal scales. Realistic forward modeling of the Sun therefore requires time-dependent three-dimensional radiation magnetohydrodynamics that account for non-equilibrium effects and, typically as a separate step, detailed radiative transfer calculations, resulting in synthetic observables that can be compared to observations. Such artificial observations sometimes also account for instrumental and seeing effects, which, in addition to aiding the interpretation of observations, provide instructive tools for designing and optimizing ALMA's solar observing modes. In the other direction, ALMA data in combination with other simultaneous observations enable the reconstruction of the solar atmospheric structure via data inversion techniques. This article highlights central aspects of the impact of ALMA for numerical modeling of the Sun and their potential and challenges, together with selected examples.