The Parker problem: existence of smooth force-free fields and coronal heating

Parker (Astrophys J 174:499, 1972) put forward a hypothesis regarding the fundamental nature of equilibrium magnetic fields in astrophysical plasmas. He proposed that if an equilibrium magnetic field is subjected to an arbitrary, small perturbation, then—under ideal plasma dynamics—the resulting magnetic field will in general not relax towards a smooth equilibrium, but rather, towards a state containing tangential magnetic field discontinuities. Even at astrophysical plasma parameters, as the singular state is approached dissipation must eventually become important, leading to the onset of rapid magnetic reconnection and energy dissipation. This topological dissipation mechanism remains a matter of debate, and is a key ingredient in the nanoflare model for coronal heating. We review the various theoretical and computational approaches that have sought to prove or disprove Parker’s hypothesis. We describe the hypothesis in the context of coronal heating, and discuss different approaches that have been taken to investigating whether braiding of magnetic field lines is responsible for maintaining the observed coronal temperatures. We discuss the many advances that have been made, and highlight outstanding open questions.