Energy-crossing number relations for braided magnetic fields

This Letter derives lower bounds on the energy of braided magnetic fields, based on crossing number techniques pioneered by Freedman and He. Mean crossing numbers are defined for magnetic fields inside a cylinder; for configurations with a uniform axial component the square of the crossing number provides a lower bound for the free magnetic energy. An energy-crossing number relation is also derived for a random field with Gaussian amplitude distribution, in the limit of small correlation lengths. These results provide information on the structure of solar coronal loops and on the problem of coronal heating.