Coiling and Squeezing: Properties of the Local Transverse Deviations of Magnetic Field Lines

Abstract

We study the properties of the local transverse deviations of magnetic field lines at a fixed moment in time. Those deviations "evolve" smoothly in a plane normal to the field-line direction as one moves that plane along the field line. Since the evolution can be described by a planar flow in the normal plane, we derive most of our results in the context of a toy model for planar fluid flow. We then generalize our results to include the effects of field-line curvature. We show that the type of flow is determined by the two non-zero eigenvalues of the gradient of the normalized magnetic field. The eigenvalue difference quantifies the local rate of squeezing or coiling of neighboring field lines, which we relate to standard notions of fluid vorticity and shear. The resulting squeezing rate can be used in the detection of null points, hyperbolic flux tubes and current sheets. Once integrated along field lines, that rate gives a squeeze factor, which is an approximation to the squashing factor, which is usually employed in locating quasi-separatrix layers (QSLs), which are possible sites for magnetic reconnection. Unlike the squeeze factor, the squashing factor can miss QSLs for which field lines are squeezed and then unsqueezed. In that regard, the squeeze factor is a better proxy for locating QSLs than the squashing factor. In another application of our analysis, we construct an approximation to the local rate of twist of neighboring field lines, which we refer to as the coiling rate. That rate can be integrated along a field line to give a coiling number, $\mathrm{N_c}$. We show that unlike the standard local twist number, $\mathrm{N_c}$ gives an unbiased approximation to the number of twists neighboring field lines make around one another. $\mathrm{N_c}$ can be useful for the study of flux rope instabilities, such as the kink instability, and can be used in the detection of flux ropes.