Why are flare ribbons associated with the spines of magnetic null points generically elongated?

Abstract: Coronal magnetic null points exist in abundance as demonstrated by extrapolations of the coronal field, and have been inferred to be important for a broad range of energetic events. These null points and their associated separatrix and spine field lines represent discontinuities of the field line mapping, making them preferential locations for reconnection. This field line mapping also exhibits strong gradients adjacent to the separatrix (fan) and spine field lines, that can be analysed using the squashing factor', $Q$. In this paper we make a detailed analysis of the distribution of $Q$ in the presence of magnetic nulls. While $Q$ is formally infinite on both the spine and fan of the null, the decay of $Q$ away from these structures is shown in general to depend strongly on the null-point structure. For the generic case of a non-radially-symmetric null, $Q$ decays most slowly away from the spine/fan in the direction in which $|{\bf B}|$ increases most slowly. In particular, this demonstrates that the extended, elliptical high-$Q$ halo around the spine footpoints observed by Masson et al. (Astrophys. J., 700, 559, 2009) is a generic feature. This extension of the $Q$ halos around the spine/fan footpoints is important for diagnosing the regions of the photosphere that are magnetically connected to any current layer that forms at the null. In light of this, we discuss how our results can be used to interpret the geometry of observed flare ribbons incircular ribbon flares', in which typically a coronal null is implicated. We conclude that both the physics in the vicinity of the null and how this is related to the extension of $Q$ away from the spine/fan can be used in tandem to understand observational signatures of reconnection at coronal null points.