Knot Probability of Random Magnetic Field Lines

In this paper, we apply several latest results from statistical physics on the probability and energy of knotting to study the knotted field lines in solar corona. Since the solar magnetic field in small scale can be seen as nearly random, by assuming that the magnetic field lines behave similarly to random loops, we find the probability P of certain knot type K for the field line knotting as a function to the distance L between the foot-points of sunspots, which is PK(L)=CKL2αKexp(−L2β). From the equation, we find that the variety of knot type increases with the distance. Since knotting is the topological resemblance to magnetic helicity, which is an invariant for ideal MHD, our result enriches the understanding of the probability of magnetic helicity as well as field line structure in active regions. Based on the relation between knotting and magnetic energy, we provide support to the high variety of field line knot types.