Taylor’s Relaxation in an Unbounded Domain of Space

Abstract

We discuss the problem of the existence of a minimum energy state among all the magnetic fields: i) occupying some unbounded domain of space D (which may be either the exterior of a bounded region, or a half-space, or a half-cylinder); ii) having a given normal component on the boundary of D; iii) having a prescribed relative helicity. We show that, depending on the shape of D and on the values of the parameters, this problem has either a unique solution (which is the unique constant-a force-free field satisfying the constraints), or no solutions at all (“minimizing sequences” losing, when passing to the limit, a part of their energy and helicity, which gets diluted in the infinite volume of D). This result is used in particular to discuss some recent attempts to extend to astrophysical systems (e.g the coronae of stars and of accretion disks) Taylor’s theory describing the turbulent relaxation of a plasma in some laboratory devices.