On the Magnetic Squashing Factor and the Lie Transport of Tangents

Abstract

The squashing factor (or squashing degree) of a vector field is a quantitative measure of the deformation of the field line mapping between two surfaces. In the context of solar magnetic fields, it is often used to identify gradients in the mapping of elementary magnetic flux tubes between various flux domains. Regions where these gradients in the mapping are large are referred to as quasi-separatrix layers (QSLs), and are a continuous extension of separators and separatrix surfaces. These QSLs are observed to be potential sites for the formation of strong electric currents, and are therefore important for the study of magnetic reconnection in three dimensions. Since the squashing factor, Q, is defined in terms of the Jacobian of the field line mapping, it is most often calculated by first determining the mapping between two surfaces (or some approximation of it) and then numerically differentiating. Tassev & Savcheva have introduced an alternative method, in which they parameterize the change in separation between adjacent field lines, and then integrate along individual field lines to get an estimate of the Jacobian without the need to numerically differentiate the mapping itself. But while their method offers certain computational advantages, it is formulated on a perturbative description of the field line trajectory, and the accuracy of this method is not entirely clear. Here we show, through an alternative derivation, that this integral formulation is, in principle, exact. We then demonstrate the result in the case of a linear, 3D magnetic null, which allows for an exact analytical description and direct comparison to numerical estimates.

The Dynamic Formation of Pseudostreamers

“…In order to analyze our simulation results, we extracted representative snapshots of the magnetic field for the SN and TN configurations, which are indicated with numeric subscripts (e.g., SN 3 ) for the times listed in Table 2. We then applied the QSLsquasher tool developed by Tassev & Savcheva (2017; see also Scott et al 2017) to create surface and volume renderings of the slog 10 Q ⊥ . We also calculated the locations of the null points in each snapshot using a routine developed by F. Chiti at the University of Dundee and based on the method of Haynes & Parnell (2007).…”

Formation and Eruption of a Mini-sigmoid Originating in Coronal Hole

“…It is found that the sigmoidal field has a highest twist of 0.8, corresponding to the erupting threads that distribute in both northern and southern parts of the minisigmiod. We calculate the squashing factor Q, which measures the change of magnetic field connectivity (Demoulin et al 1996;Titov et al 2002;Titov 2007), following the method of Tassev & Savcheva (2017) and Scott et al (2017). Similar to what is revealed in observed EUV images, the distribution of the Q value also presents two groups of sigmoidal structures.…”

Definition of the Spatial Propagator and Implications for Magnetic Field Properties

“…Equation 22 with initial condition 23 , generalizes the field-line deviation to include affine transformations in the connectivity parameter representations (see, e.g. , Tassev and Sevcheva, 2017 ; Scott, Pontin, and Hornig, 2017 , for application).…”

“…The QSL is often described in terms of the “squashing” of a flux tube (see, e.g. , Titov, 2007 ; Tassev and Sevcheva, 2017 ; Scott, Pontin, and Hornig, 2017 ); essentially, a relational measure between the eccentricity of the flux-tube cross-sectional area at the system boundaries, and hence is inherently 2D. Moreover, the squashing of a flux tube is defined entirely independent of rotation.…”

“…The field lines define a map from -coordinates with origin at , to -coordinates with origin at . By Equations 16 and 17 , the basis vectors are propagated, via Lie transport (Scott, Pontin, and Hornig, 2017 ), from to . The matrix may be interpreted as the Jacobian of this map, given by the corresponding components of the matrix representation of the Spatial Propagator .…”