The behaviour of neutral lines in two-dimensional magnetohydrodynamic equilibria

Nonlinear equilibrium solutions for two-dimensional magnetic arcades (partial derivative/partial derivative z = 0) using a Grad-Shafranov equation in which the axial magnetic field and the pressure are specified as functions of the component of the vector potential in the z direction are re-examined. To compute nonlinear solutions one is restricted to seeking solutions on finite computational domains with specified boundary conditions. We consider two basic models which have appeared in the literature. In one model the field is laterally restricted by means of Dirichlet boundary conditions and free to extend vertically by means of a Neumann condition at the top of the domain. Far such fields, bifurcating solutions only appear for a narrow range of values for the parameter lambda (the ratio of a typical length scale of the held to the gravitational scale height). Nevertheless, we show that the presence of this parameter is essential for bifurcating solutions in such domains. For the second model with Neumann conditions on three sides of the domain representing the region above the photosphere we do not find bifurcating solutions. Instead high-energy solutions with detached field lines evolve smoothly from low-energy solutions which have all field lines attached to the photosphere. Again the presence or absence of detached flux is dependent on the magnitude of lambda for those fields which are evolved quasi-statically via an increase in the plasma pressure.