# Calculate Squashing Factor and Twist Number

Rui Liu and Jun Chen

We have developed a computationally fast code to calculate the squashing factor $$Q$$ and the twist number $$\mathcal{T}_w$$, introduced in Liu et al. 2016, ApJ, 818, 148 (arXiv: 1512.02338).

Note: IDL procedures included in SolarSoft Ware are not specified here.
ProcedureDescription
qfactor.proCalculate the squashing factor Q and twist number Tw at the photosphere or at a cross section or in a box volume, given a 3D magnetic field with uniform grids in Cartesian coordinates, calling the Fortran module qfactor.f90 (version 2016.06.22)
check_slash.proAdd delimiter to the end of directory paths
doppler_color.proLoad a color table of blue --> red
doppler_color_mix.proLoad a color table of blue -> red -> green -> yellow
rev_true_image.proreform a true-color image of (m x n x 3) to (3 x m x n), as required by IDL's write_png
sign2d.proReturn the signs (+/-) of each element of a 2D array
qfactor.f90Fortran module to do the calculation; If compiled by ifort: ifort -o qfactor qfactor.f90 -parallel -openmp -mcmodel=large
trace_bline.f90Fortran module to do the field line tracing

As of Jan 25, 2022, an optimized version is available below

ProcedureDescription
qfactor.proSimilar to the previous version
qfactor.f90Fortran module to do the calculation; Can be compiled either by ifort or gfortran: ifort -o qfactor.x qfactor.f90 -fopenmp -mcmodel=large -O3 -xHost -ipo; gfortran -o qfactor.x qfactor.f90 -fopenmp -mcmodel=large -O3
trace_bline.f90Fortran module to do the field line tracing. In stead of the classic RK4 as in the previous version, RKF45 (Fehlberg 1969) is now adopted for further optimization
trace_scott.f90Fortran module to do the field line tracing following Scott et al. 2017, ApJ, 848, 117
doppler_color.proLoad a color table of blue --> red, same as the previous version

## Definition of $$Q$$

For a mapping defined by the two footpoints of a field line $$\Pi_{12}: \mathbf{r}_1(x_1,\ y_1)\mapsto \mathbf{r}_2(x_2,\ y_2),$$ the Jacobian matrix associated with the mapping is $D_{12}=\left[\frac{\partial \mathbf{r}_2}{\partial \mathbf{r}_1}\right]= \begin{pmatrix} \partial x_2/\partial x_1 &\partial x_2/\partial y_1 \\ \partial y_2/\partial x_1 & \partial y_2/\partial y_1 \end{pmatrix} \equiv \begin{pmatrix} a & b \\ c & d \end{pmatrix},$ and then the squashing factor associated with the field line is defined as (see Titov et al. 2002) $Q\equiv \frac{a^2+b^2+c^2+d^2}{|B_{n,1}(x_1,y_1)/B_{n,2}(x_2,y_2)|},$ where $$B_{n,1}(x_1,y_1)$$ and $$B_{n,2}(x_2,y_2)$$ are the components normal to the plane of the footpoints, in our case, the photosphere, and their ratio is equivalent to the determinant of $$D_{12}$$.

## Definition of $$\mathcal{T}_w$$

$$\mathcal{T}_w$$ can be used to measure how many turns two infinitesimally close field lines wind about each other (see Eq. (16) in Berger and Prior 2006) $\begin{eqnarray} \mathcal{T}_w&=&\int_L\frac{\mu_0J_\parallel}{4\pi |B|}\,dl = \int_L\frac{\nabla\times\mathbf{B}\cdot\mathbf{B}}{4\pi B^2}\,dl \label{eq:Tw} \\ &=&\frac{1}{4\pi}\int_L \alpha\,dl, \quad \text{if } \nabla\times\mathbf{B}=\alpha \mathbf{B}, \label{eq:Tw_alpha} \end{eqnarray}$ where $$\alpha$$ is the force-free parameter, and $$\nabla\times\mathbf{B}\cdot\mathbf{B}/4\pi B^2$$ can be regarded as a local density of twist along the field line.