Spatially Periodic, Force-free Magnetic Fields with Resistive Decay*

Force-free magnetic fields, i.e. fields that are either parallel or antiparallel to the electric current, occur in the solar chromosphere and also have been used in certain magnetic confinement schemes in controlled thermonuclear fusion research. In this paper, we derive a general expression for force-free fields that decay resistively and are spatially periodic. The general expression that is found consists of a sum of spatial Fourier modes with the property that all wave vectors have the same absolute magnitude, with components that satisfy a Diophantine equation. A large class of particular solutions is found to exist. Jette's theorem in resistive magnetohydrostatics is rederived for the case of periodic fields by means of a simple geometrical proof. In the concluding section the results are extended to the case where the electric conductivity is a function of time. An application to the study of iso-energetic Beltrami flows in classical fluid mechanics is indicated.