Understanding the connection between solar atmospheric layers through the magnetic field is a challenging task in solar physics. These regions are form by plasma weakly ionized and in some places the ionization fraction can be as low as 10^-4 meaning that the presence of neutrals could be important. The interaction of weakly ionized plasma with magnetic fields entails the arise of non-ideal effects and the departure of the ideal magnetohydrodynamic (MHD) framework. Non-ideal effects are Ohmic diffusion, ambipolar diffusion, the Hall effect, and the Biermann battery effect, being the ambipolar diffusion the only one directly related to the presence of neutrals. These effects appear as a series of extra terms into the generalized Ohm’s law and into the total energy equation, conforming this way the non-ideal MHD set of equations. Under certain conditions, these terms could introduce severe restrictions for solving numerically the equations, for example on the integration time step or compromising the stability of the numerical scheme.
In this thesis, two numerical schemes are introduced into the MHD numerical code Mancha3D to overcome those limitations. The first of them is known as super time-stepping (STS) and it is designed to overcome the limitations imposed over the temporal time step by parabolic equations and can be applied to the ambipolar diffusion term. The second scheme is called the Hall diffusion scheme (HDS) and it is used when the Hall term becomes dominant. These two numerical techniques can be used together by using the Strang operator splitting technique. The validation for each of these schemes is provided by comparing the analytical solution with the numerical one for a suite of numerical tests.
Next, using the new code, a set of numerical experiments are performed to study the wave mode transformation by means of the Hall term into a quasi-realistic stratification in thermodynamic parameters resembling the solar atmosphere. The results confirm the efficacy of the mechanism for the solar case. The efficiency of mode transformation is a sensitive function of the angle between the wave propagation direction and the magnetic field, and of the wave frequency. It also increases when the field direction and the wave direction are aligned for increasing wave frequencies.
Finally realistic three-dimensional simulations of solar local dynamo where the magnetic field is seeded by the battery effect are performed. This magnetic field is amplified to solar values by the convective local dynamo. Ambipolar effect allows to dissipate incompressible perturbations associated to magnetic waves removing the magnetic Poynting flux and converting the magnetic energy into the thermal energy. Hall term introduces a new fast-to-Alfvén mode transformation mechanism and helps generating currents. The Alfvén waves generated could travel to upper layers and the currents could be dissipated by the ambipolar diffusion closing the cycle. The action of all three effects can be considered as an attractive mechanism of energy generation, transport, and chromospheric heating.