# 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 and 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 onmore »

- Authors:

- University of Dundee Nethergate, Dundee (United Kingdom)

- Publication Date:

- OSTI Identifier:
- 22679747

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Astrophysical Journal; Journal Volume: 848; Journal Issue: 2; Other Information: Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 79 ASTROPHYSICS, COSMOLOGY AND ASTRONOMY; ACCURACY; APPROXIMATIONS; COMPARATIVE EVALUATIONS; DEFORMATION; LAYERS; MAGNETIC FIELDS; MAGNETIC FLUX; MAGNETIC RECONNECTION; SUN; SURFACES; TRAJECTORIES; VECTOR FIELDS

### Citation Formats

```
Scott, Roger B., Pontin, David I., and Hornig, Gunnar.
```*On the Magnetic Squashing Factor and the Lie Transport of Tangents*. United States: N. p., 2017.
Web. doi:10.3847/1538-4357/AA8A64.

```
Scott, Roger B., Pontin, David I., & Hornig, Gunnar.
```*On the Magnetic Squashing Factor and the Lie Transport of Tangents*. United States. doi:10.3847/1538-4357/AA8A64.

```
Scott, Roger B., Pontin, David I., and Hornig, Gunnar. Fri .
"On the Magnetic Squashing Factor and the Lie Transport of Tangents". United States.
doi:10.3847/1538-4357/AA8A64.
```

```
@article{osti_22679747,
```

title = {On the Magnetic Squashing Factor and the Lie Transport of Tangents},

author = {Scott, Roger B. and Pontin, David I. and Hornig, Gunnar},

abstractNote = {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 and 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.},

doi = {10.3847/1538-4357/AA8A64},

journal = {Astrophysical Journal},

number = 2,

volume = 848,

place = {United States},

year = {Fri Oct 20 00:00:00 EDT 2017},

month = {Fri Oct 20 00:00:00 EDT 2017}

}