"The broad scientific objective of the SPIA proposal is to understand the magnetism of the Sun and stars and to establish connections between the magnetic activity in sub-surface layers and its manifestation in the outer atmosphere. The complex interactions in magnetized stellar plasmas are best studied via numerical simulations, a new powerful method of research that appeared in astrophysics with the development of large supercomputer facilities. With a coming era of large aperture solar telescopes, ATST and EST, spectropolarimetric observations of the Sun will become available at extraordinary high spatial and temporal resolutions. New modelling tools are required to understand the plasma behaviour at these scales. I propose to consolidate a research group of bright scientists around the PI to explore a novel promising approach for the description solar atmospheric plasma under multi-fluid approximation. The degree of plasma ionization in the photosphere and chromosphere of the Sun is extremely low and significant deviations from the classical magneto-hydrodynamic description are expected. A major development of the SPIA proposal will be the implementation of a multi-fluid plasma description, appropriate for a partially ionized medium, relaxing approximations of classical magneto-hydrodynamics. With the inclusion of standard radiative transfer into the three-dimensional multi-fluid code to be developed by the project team, it will be possible to perform simulations of solar sub-photospheric and photospheric regions, up to the low chromosphere, with a realism not achieved before. The importance of the non-ideal plasma effect for the energy balance of the solar chromosphere will be evaluated, and three-dimensional time-dependent models of multi-fluid magneto-convection will be created. This effort will produce a significant step toward the solution of the long-standing question of the origin of solar chromosphere, one of the most poorly understood regions of the Sun."