Estimation of the squashing degree within a three-dimensional domain

Context. The study of the magnetic topology of magnetic fields aims at determining the key sites for the development of magnetic reconnection. Quasi-separatrix layers (QSLs), regions of strong connectivity gradients, are topological structures where intense-electric currents preferentially build-up, and where, later on, magnetic reconnection occurs.
Aims: QSLs are volumes of intense squashing degree, Q; the field-line invariant quantifying the deformation of elementary flux tubes. QSL are complex and thin three-dimensional (3D) structures difficult to visualize directly. Therefore Q maps, i.e. 2D cuts of the 3D magnetic domain, are a more and more common features used to study QSLs.
Methods: We analyze several methods to derive 2D Q maps and discuss their analytical and numerical properties. These methods can also be used to compute Q within the 3D domain.
Results: We demonstrate that while analytically equivalent, the numerical implementation of these methods can be significantly different. We derive the analytical formula and the best numerical methodology that should be used to compute Q inside the 3D domain. We illustrate this method with two twisted magnetic configurations: a theoretical case and a non-linear force free configuration derived from observations.
Conclusions: The representation of QSL through 2D planar cuts is an efficient procedure to derive the geometry of these structures and to relate them with other quantities, e.g. electric currents and plasma flows. It will enforce a more direct comparison of the role of QSL in magnetic reconnection.