Steady-state axisymmetric nonlinear magnetohydrodynamic solutions with various boundary conditions

Axisymmetric magnetohydrodynamics (MHD) can be invoked for describing astrophysical magnetized flows and formulated to model stellar magnetospheres including main-sequence stars (e.g. the Sun), compact stellar objects [e.g. magnetic white dwarfs (MWDs), radio pulsars, anomalous X-ray pulsars, magnetars, isolated neutron stars, etc.] and planets as a major step forward towards a full three-dimensional model construction. Using powerful and reliable numerical solvers based on two distinct finite-difference method and finite-element method schemes of algorithm, we examine axisymmetric steady-state or stationary MHD models in Throumoulopoulos & Tasso, finding that their separable semi-analytic non-linear solutions are actually not unique given their specific selection of several free functionals and chosen boundary conditions. Similar situations of multiple non-linear solutions with the same boundary conditions actually also happen to force-free magnetic field models of Low & Lou. The multiplicity of non-linear steady MHD solutions gives rise to differences in the total energies contained in the magnetic fields and flow velocity fields as well as in the asymptotic behaviours approaching infinity, which may in turn explain why numerical solvers tend to converge to a non-linear solution with a lower energy than the corresponding separable semi-analytic one. By properly adjusting model parameters, we invoke semi-analytic and numerical solutions to describe different kinds of scenarios, including nearly parallel case and the situation in which the misalignment between the plasma flow and magnetic field is considerable. We propose that these MHD models are capable of describing the magnetospheres of MWDs as examples of applications with moderate conditions (including magnetic field) where the typical values of several important parameters are consistent with observations. Physical parameters can also be estimated based on such MHD models directly. We discuss the challenges of developing numerical extrapolation MHD codes in view of the non-linear degeneracy.