Constrained optimization approach for magnetohydrostatic equilibria on the solar atmosphere

Context. The magnetic field plays an essential role in the evolution of structures and the description of events in the solar atmosphere. Several models have been developed to reconstruct the magnetic field, due to the impossibility of its direct measurement in the solar corona. The model proposed here extrapolates the photospheric magnetogram data up to the corona using a constrained optimization method. In the upper photosphere and chromosphere, both the magnetic and nonmagnetic forces must be taken into account, and the magnetic field reconstruction must be done considering the plasma pressure and density. This is done by applying the Lagrange multiplier technique, as the constrained optimization method, to compute the magnetic field, plasma pressure, and density in magnetohydrostatic equilibria. Aims. This approach has previously been introduced to reconstruct a nonlinear force-free magnetic field. For this work we extended it to a more realistic issue to reconstruct the magnetic field and calculate the plasma pressure and density in a magnetohydrostatic environment. Methods. Our approach was to use the constrained optimization method, which is computationally more efficient and easy to implement. The Lagrange multiplier technique is a powerful mathematical tool that has been successfully applied to many areas of physics. We sought to minimize a Lagrangian, which minimizes the divergence term subject to the constraint magnetohydrostatic equilibrium equation. The plasma parameters and magnetic field were eventually computed following the iteration scheme along with appropriate boundary data. Results. In our previous work, we applied Lagrange multiplier techniques to reconstruct a force-free magnetic field for the solar atmosphere. For this wok, we extended the same optimization technique to extrapolate magnetic field and plasma parameters in magnetohydrostatic equilibria. The results for the magnetic field and plasma parameters were calculated and compared with those obtained by other models in the magnetohydrostatic equilibrium environment as well as the semi-analytical solution as a reference model. Conclusions. A force-free magnetic field and a suitable distribution for pressure and density were used as the initial input for their corresponding evolution equations. After 20 000 iterations, the convergence of the Lagrangian in our model was slightly better than that of the comparison model. The indicators such as the relative magnetic energy and magnetic field lines were investigated, which are in agreement with the reference model compared to the comparison model.