Self-consistent generation of the ambipolar electric field in collisionless plasmas via multi-mode electrostatics

In this work, we investigate the generation of the ambipolar electric field in a gravitationally stratified, collisionless plasma atmosphere. In such environments, gravity tends to separate charged species. To prevent separation an electric field, classically described by the Pannekoek-Rosseland expression, is usually imposed externally. Here, we propose a self-consistent method to recover this field based on a multi-mode Fourier expansion of the electrostatic interaction. We show that, under suitable conditions, this approach naturally leads to the ambipolar electric field and restores charge neutrality. The method is tested in both isothermal and multi-temperature plasma configurations. This framework provides a foundation for future developments that may include collisions, ionization, and asymmetric boundary conditions to model more realistic stellar atmospheres.