Three-dimensional force-free looplike magnetohydrodynamic equilibria

Computations of three-dimensional force-free magnetohydrodynamic (MHD) equilibria, del x B = lambdaB with lambda = lambda₀, a constant are presented. These equilibria are determined by boundary conditions on a surface corresponding to the solar photosphere. The specific boundary conditions used correspond to looplike magnetic fields in the corona. It is found that as lambda₀ is increased, the loops of flux become kinked, and for sufficiently large lambda₀, develop knots. The relationship between the kinking and knotting properties of these equilibria and the presence of a kink instability and related loss of equilibrium is explored. Clearly, magnetic reconnection must be involved for an unknotted loop equilibrium to become knotted, and speculations are made about the creation of a closed hyperbolic field line (X-line) about which this reconnection creating knotted field lines is centered.