# Definition of the Spatial Propagator and Implications for Magnetic Field Properties

## Abstract

We present a theoretical framework to analyze the 3D coronal vector magnetic-field structure. We assume that the vector magnetic field exists and is *a priori* smooth. We introduce a generalized connectivity phase space associated with the vector magnetic field in which the basic elements are the *field line* and its linearized variation: the *Spatial Propagator*. We provide a direct formulation of these elements in terms of the vector magnetic field and its spatial derivatives, constructed with respect to general curvilinear coordinates and the equivalence class of general affine parameterizations. The Spatial Propagator describes the geometric organization of the local bundle of field lines, equivalent to the kinematic deformation of a propagated volume tied to the bundle. The Spatial Propagator’s geometric properties are characterized by dilation, anisotropic stretch, and rotation. Extreme singular values of the Spatial Propagator describe quasi-separatrix layers (QSLs), while true separatrix surfaces and separator lines are identified by the vanishing of one and two singular values, respectively. Finally, we show that, among other possible applications, the squashing factor [\(Q\)] is easily constructed from an analysis of particular sub-matrices of the Spatial Propagator.

## Keywords

Magnetic fields, corona Corona, structures## 1 Introduction

Geometry describes the measurable lengths and angles associated with a system’s configuration, whereas topology is concerned with those properties preserved under continuous deformation. Over the past several decades the importance of the geometric and topological features to fluid and plasma dynamics has become increasingly clear (see, *e.g.*, Moffatt *et al.*, 1991; Arnold and Khesin, 1998; Ricca, 2001). Algebraic and geometric analyses of magnetic and hydrodynamic structures have provided significant development in the context of general plasma equilibria (*e.g.* Moffatt, 1985, 1986), stability (*e.g.* Bulanov and Sasorov, 1978; Syrovatskii, 1981; Bulanov *et al.*, 1999), reconnection (Hesse and Schindler, 1988; Schindler, Hesse, and Birn, 1988; Ruzmaikin and Akhmetiev, 1994; Bulanov *et al.*, 2002; Pontin *et al.*, 2005), heating, and wave generation (Ruzmaikin and Berger, 1998). In particular, geometric and topological analyses of the solar coronal magnetic field have focused on understanding heating and dynamics (Antiochos, 1987; Berger, 1994; Priest, Longcope, and Heyvaerts, 2005), as well as investigations into the eruptive phenomenology of flares (Mandrini *et al.*, 1995; Démoulin *et al.*, 1997; Titov and Démoulin, 1999; Aulanier *et al.*, 2000) and coronal mass ejection (CME) initiation (Antiochos, DeVore, and Klimchuck, 1999; Lynch *et al.*, 2008; Lynch and Edmondson, 2013).

The build-up, storage, transport, and subsequent release of magnetic energy in the low-\(\beta \) solar corona is widely accepted as the basic requirement for solar coronal heating, the origin and generation of the solar wind, as well as eruptive phenomenology and space-weather prediction (see, *e.g.*, Klimchuk, 2006, for a review). It is the geometric and topological features of the coronal magnetic field that govern these dynamics in the low-\(\beta \) coronal plasma environment (see Longcope, 2005, for a review). In general, non-trivial geometric and topological features such as null-point structure (Lau and Finn, 1990; Parnell *et al.*, 1996), separator lines (Longcope and Cowley, 1996), bald patches (Seehafer, 1986; Wolfson, 1989; Low, 1992; Titov, Priest, and Démoulin, 1993), separatrix surfaces (Low, 1987; Somov, 1992; Priest, Heyvaerts, and Title, 2002), and quasi-separatrix layers (QSLs: Priest and Démoulin, 1995; Démoulin *et al.*, 1996, 1997; Titov, 1999; Milano *et al.*, 1999) appear ubiquitously in the coronal magnetic field. In fact, the current state of global solar-wind generation models (see, *e.g.*, Abbo *et al.*, 2016, for a recent review), differ in the extent and complexity of these geometric and topological structures within the coronal magnetic field (*e.g.* Wang and Sheeley, 1990; Fisk, Schwadron, and Zurbuchen, 1998; Arge and Pizzo, 2000; Fisk, 2003; Cranmer, van Ballegooijen, and Edgar, 2007; Antiochos *et al.*, 2011; Antiochos, 2013).

Since the vector magnetic field is reference-frame dependent, so too is the magnetic-connectivity description (Hornig and Schindler, 1996); in fact, the entire concept of a magnetic-field line is not Lorentz invariant (Hornig, 1997). However, in a fixed reference frame, the geometric and topological features of the magnetic field constrain the dynamics. In general, the geometric features and topological constraints of the magnetic field are primarily important to understand where and how energy is stored and released in low-\(\beta \) plasma environments such as the solar corona. In the presence of resistive (non-ideal) physics, field-line connectivity topology is no longer preserved, although separatrix structures remain. Separatrix surfaces and QSLs are associated with electric-current-sheet formation and reconnection, and therefore it is the geometry of these structures that constrain the storage of magnetic energy throughout the coronal volume, and determine both the location and the ability to release free energy (see the references above). In particular, QSL locations are related to observations of sudden flare brightening in H\(\alpha \) for a wide variety of solar-flare phenomenology such as circular ribbon flares, two ribbon flares, and twisted flux rope (sigmoidal active region) morphologies (see, *e.g.*, Janvier, 2017, and the references therein). Furthermore, separatrix surfaces and QSLs are the boundaries dividing regions of disparate connectivity in complex, multi-polar coronal structures. Hence, QSLs feature prominently in the rapid reorganization of the vector magnetic field and subsequent energy release in CME initiation mechanisms (see, *e.g.*, Aulanier, Démoulin, and Grappin, 2005; Effenberger *et al.*, 2011; Janvier *et al.*, 2014; Schmieder, Aulanier, and Vršnak, 2015; Lynch *et al.*, 2016, and the references therein).

The features of the magnetic connectivity map such as helicity, separatrix and QSL structures, and their effect on the system dynamics find their most explicit representations in terms of the algebraic and geometric descriptions of the field line and the field-line bundle. In other words, the constraints on the system dynamics become transparent when cast in terms of the geometric structure and topological invariants of the field-line bundle and its behavior. Numerical magnetohydrodynamic (MHD) methodologies allow one to explore the locus of system dynamics for various magnetic-field models, thermodynamic heating models, and boundary conditions. While the MHD method (where applicable) is correct, the MHD equations and numerical solutions are often opaque to these explicit geometric structures (*e.g.* QSLs, separatrices, and separators) and their behavior. Moreover, many numerical difference schemes employ linear interpolation, which has the potential to limit these codes as an accurate representation of the system, especially in the vicinity of sharp gradients and other small-scale structures. Effectively, linear interpolation shifts the problem of the small-scale dynamics from the physical quantities on to finer and more complex numerical-grid resolutions. This is an extremely popular approach to numerical/computational solar- and space-plasma physics, but arguably leads to spurious effects (see, *e.g.*, Edmondson, 2012, for a discussion); we offer no judgment regarding the veracity of these approaches. This perspective, however, requires the development of new mathematical tools/description/framework to analyze the geometric organization of the magnetic-field connectivity map.

- i)
the vector magnetic field is the

*primary (observable)*quantity that satisfies some standard physical evolutions (*e.g.*Maxwell’s equations, MHD induction,*etc.*), and the connectivity map is the*secondary (derived)*quantity; and - ii)
while large gradients may exist, the vector magnetic field is

*a priori*smooth everywhere.

*Spatial Propagator*, from the linearized variation of a field line. We demonstrate that the Spatial Propagator characterizes the geometric organization of a local bundle of field lines. Moreover, we identify topological invariants derived from the Spatial Propagator, as well as demonstrate the proper limiting connection to QSLs, separatrix surfaces, and separator lines. Beyond the limiting cases, the inclusion and analysis of existing and/or generated singular structures within the vector field are outside the scope of the present work.

The roadmap for this article is as follows: In Section 2 we lay out a precise mathematical definition for the field lines of a vector field (Section 2.1). We introduce the Spatial Propagator (Section 2.2) as a generalized spatial variation of an entire field-line solution, which describes the local bundle of field lines. Furthermore, we derive a direct relation between the Spatial Propagator and the local gradient of the vector field (Section 2.3), and hence the precise mathematical formulation of the geometric phase space, consisting of the integral curve solutions and their associated Spatial Propagators. Finally, we discuss the various representations of the Spatial Propagator (Section 2.4): covariance with respect to local curvilinear coordinates, and equivalence with respect to affine transformations of the coordinate defined along the field line.

In Section 3 we characterize the geometric organization (dilation, rotation, anisotropic stretch, and connectivity gradient) of a local bundle of field lines using a mathematically equivalent kinematic analysis of a volume propagated along and deformed by the bundle. We explore the vector-field geometry using the Spatial Propagator in terms of volumetric dilation (Section 3.1), from which we identify a topological invariant measure that reflects the divergence-free condition of physical magnetic fields. We demonstrate that the singular values and singular vectors of the Spatial Propagator characterize the anisotropic stretch and rigid-body rotation deformations of the geometry (Section 3.2). Lastly (Section 3.3), we identify quasi-separatrix structures directly from the Spatial Propagator as extreme kinematic deformations, and we demonstrate the construction of the \(Q\)-factor (see, *e.g.*, Titov, Hornig, and Démoulin, 2002; Titov, 2007) by a simple example.

Summary of major objects and main equations of this framework.

Symbol | Quantity name | Equations | Section |
---|---|---|---|

)r | Vector field | ||

| Connectivity parameter | ||

\(\boldsymbol{r}_{0}\) | Reference point | ||

\(\boldsymbol{r} ( \lambda , \boldsymbol{r}_{0} )\) | Field line (for single fixed \(\boldsymbol{r}_{0}\)) | ||

\(\boldsymbol{r} ( \lambda , \boldsymbol{r}_{0} )\) | Congruence (for all \(\boldsymbol{r}_{0} \in \Omega _{0} \subset \mathbb{R}^{3}\)) | ||

∇ )r | Covariant differential vector field matrix | ||

\(\mathbf{F} ( \lambda , \boldsymbol{r}_{0} )\) | Spatial Propagator | ||

\(\boldsymbol{v} ( \lambda , \boldsymbol{r}_{0} )\) | Propagated shift vector | ||

\(\boldsymbol{h} = \boldsymbol{v} ( 0 , \boldsymbol{r}_{0} )\) | Reference shift vector | ||

\({\mathrm{d}} \Omega ( \lambda , \boldsymbol{r}_{0} )\) | Signed differential volume element | ||

\(\Omega ( \lambda , \boldsymbol{r}_{0} )\) | Total propagated volume | ||

\(\sigma _{\alpha } ( \lambda , \boldsymbol{r}_{0} )\) | Singular values of \(\mathbf{F} ( \lambda , \boldsymbol{r}_{0} )\) | ||

\({\hat{\boldsymbol{l}}}_{\alpha } ( \lambda , \boldsymbol{r}_{0} )\), \({\hat{\boldsymbol{r}}}_{\alpha } ( \lambda , \boldsymbol{r}_{0} )\) | Left-, right-singular vectors of \(\mathbf{F} ( \lambda , \boldsymbol{r}_{0} )\) | ||

\(\mathbf{V} ( \lambda , \boldsymbol{r}_{0} )\), \(\mathbf{U} ( \lambda , \boldsymbol{r}_{0} )\) | Left-, right-stretch matrices | ||

\(\mathbf{R} ( \lambda , \boldsymbol{r}_{0} )\) | Rotation matrix | ||

\(Q ( \lambda , \boldsymbol{r}_{0} )\) | Squashing factor [ |

## 2 The Integral Curve Description of Vector Field Geometry

^{1}basis \(\lbrace \hat{\boldsymbol{e}}_{x}, \hat{\boldsymbol{e}}_{y}, \hat{\boldsymbol{e}}_{z} \rbrace \), the vector field \(\boldsymbol{B} \left ( \boldsymbol{r} \right )\) has component functions,

In general, the vector field \(\boldsymbol{B} \left ( \boldsymbol{r} \right )\) is time-dependent, and therefore, strictly speaking, so too are the integral curves (Section 2.1), and the Spatial Propagator (Section 2.2), as well as all other objects constructed therefrom. For ease of notation, throughout this work we suppress the functional time-dependence of all quantities; this may be interpreted as analyzing the system at a fixed time, or for very-low frequency dynamics, \(f \ll c/L\) where \(c\) is the characteristic speed of communication, and \(L\) is a characteristic length scale. The dynamical description of the integral curves (Section 2.1), the Spatial Propagator (Section 2.2), *etc.* require a treatment of the full four-dimensional electromagnetic-field tensor (Jackson, 1999, Section 11.9), which is outside the scope of this work.

### 2.1 The Integral Curves of a Vector Field

*connectivity parameter*, and a three-component vector \(\boldsymbol{r}_{0} \in M\), referred to as the

*reference point*. A single integral curve, as a particular solution to Equations 2 and 3, is identified by a single fixed reference point \(\boldsymbol{r}_{0}\). The connectivity parameter \(\lambda \) denotes the

*distance per unit field strength*along a particular solution curve issuing from a particular reference point. We note that the rate of change with respect to \(\lambda \) is simply the directional derivative along the vector field, which may be written with respect to the coordinate representation

The integral curve solutions \(\boldsymbol{r} ( \lambda , \boldsymbol{r}_{0} )\) are known by various names depending on the nature and interpretation of the differential equations and vector field. In physics, integral curves for electric and magnetic fields are known as *field lines*, whereas the integral curves for a velocity field are called *streamlines*. In general dynamical systems theory, the integral curves of the governing differential equation system are referred to as *trajectories* or *orbits*.

*field line*to denote a particular solution to Equations 2 and 3 with fixed \(\boldsymbol{r}_{0}\). Moreover, a

*flux tube*is a

*congruence of integral curves*(or simply a

*congruence*); that is a local bundle of field lines defined by a particular set of reference points in some neighborhood \(\boldsymbol{r}_{0} \in \Omega _{0} \subseteq M\) (see Figure 1).

We make a few observations and define some nomenclature regarding the field lines in physical systems of interest. The reference point \(\boldsymbol{r}_{0}\) is a free parameter, typically chosen on the system boundary, or to coincide with some known initial state within the system interior; for example, in a magnetized plasma this choice typically coincides with a highly-conductive parcel of plasma material. The \(\lambda = 0\) datum defined by the reference point \(\boldsymbol{r}_{0} = \boldsymbol{r} ( 0, \boldsymbol{r}_{0} )\) is typically referred to as the *launch footpoint* in magnetic systems. The point \(\boldsymbol{r} = \boldsymbol{r} ( L, \boldsymbol{r}_{0} )\) for a *finite*\(\lambda = L\), corresponding to the final point of integration of Equations 2 and 3 is often referred to as the *target footpoint* in magnetic systems; typically, the target footpoint is where the integral curves crosses the system boundary, or encounters a singular structure in the vector field.

^{2}and \(b \in \mathbb{R}\) an arbitrary constant.

^{3}The re-parametrized flow \(\boldsymbol{r} ( \ell , \boldsymbol{r}_{b} )\) satisfies Equations 2 and 3 for vector field \(\boldsymbol{X} ( \boldsymbol{r} ) = \boldsymbol{B} ( \boldsymbol{r} ) / f ( \boldsymbol{r} )\) and reference condition \(\boldsymbol{r}_{b}\); that is,

A simple application of this \(\lambda \mapsto \ell ( \lambda )\) re-parametrization is the analysis of the \(\boldsymbol{B}\) field in a system with boundaries \(\partial M\). One has the freedom to choose the function \(f ( \boldsymbol{r} )\) (and constant \(b = 0\)) in order that the values \(\ell = 0\) and \(\ell = 1\) coincide with the initial and final points of the field lines taken at the boundary surfaces, \(\boldsymbol{r}_{0} \in \partial M\) and \(\boldsymbol{r} ( 1, \boldsymbol{r}_{0} ) \in \partial M\); typically at the photosphere for solar coronal applications. With \(\ell = 0\) and \(\ell = 1\) as boundary values, the connectivity parameter \(\ell \) is not the physical (dimensional) arc length, but rather a normalized dimensionless arc-length parameter.

*arc-length representation*.

### 2.2 The Spatial Propagator: A Local Congruence of Integral Curves

The relative geometry of a local congruence (*i.e.* dilation, anisotropic stretch, and rotation of the bundle of curves) is completely described by examining a field line under a spatial variation of the reference point \(\boldsymbol{r}_{0} \mapsto \boldsymbol{r}_{0} + \boldsymbol{h}\). We denote the spatial variation of the reference point by the *reference shift vector*\(\boldsymbol{h}\).

*entire*field line. To see this, consider two spatially neighboring field-line solutions within the congruence, respectively: \(\boldsymbol{r} ( \lambda , \boldsymbol{r}_{0} )\) and \(\boldsymbol{r} ( \lambda , \boldsymbol{r}_{0} + \boldsymbol{h} )\). We may relate the component functions of the neighboring field lines by a Taylor expansion, such that for all \(\lambda \) and \(\vert \boldsymbol{h} \vert \ll h_{m}\),

Equation 10 describes the local organization of all field lines with reference points within an initial volume \(\Omega _{0}\) of characteristic size \(\vert \Omega _{0} \vert ^{1/3} \ll h_{m}\); that is to say, the congruence is *local* with respect to reference points \(\boldsymbol{r}_{0} + \boldsymbol{h} \in \Omega _{0}\) with \(\vert \boldsymbol{h} \vert \ll h_{m}\).

We remark, there is an implicit assumption in Equation 10 that the magnetic vector field \(\boldsymbol{B} ( \boldsymbol{r} )\) is described by smooth component functions everywhere within the domain. In future work, we will explore the consequences of relaxing this assumption.

*Spatial Propagator*\(\mathbf{F} ( \lambda , \boldsymbol{r} _{0} )\). Then for all \(\lambda \), the variational derivative may be represented as a \(3 \times 3\) matrix whose component functions are simply the derivatives of Equation 5 with respect to the reference point components,

*The matrix representation of the Spatial Propagator*\(\mathbf{F} ( \lambda , \boldsymbol{r}_{0} )\)*contains all of the geometric information of the local congruence; that is, it carries the local geometric organization of the field lines within a neighborhood*\(\Omega _{0} \ll h_{m}^{3}\)*of a particular field-line solution*\(\boldsymbol{r} ( \lambda , \boldsymbol{r}_{0} )\). We give a precise meaning to this statement in Section 3.

### 2.3 Direct Relation Between the Spatial Propagator and the Vector Field

Like the field lines of the congruence, the Spatial Propagator \(\mathbf{F} ( \lambda , \boldsymbol{r}_{0} )\) is a function of both the connectivity parameter \(\lambda \) and reference point \(\boldsymbol{r}_{0}\). In order to calculate the Spatial Propagator, one may integrate *all* field lines, and then construct the difference Equation 10 between any two neighboring field lines, taking care to evaluate the scale length \(h_{m}\) for every field line. However, in typical physical systems of interest the governing equations describe the evolution of the vector field (*e.g.* Faraday’s law, MHD induction, *etc.*), and the associated field lines are derived therefrom. Hence, we seek a formulation of the Spatial Propagator directly from the vector field, which allows the simultaneous calculation of the spatial behavior of *all* field lines within the \(h_{m}\) neighborhood.

*e.g.*, Kobayashi and Nomizu, 1963, p. 29) of the propagated shift vector \(\boldsymbol{v}\) along the vector field \(\boldsymbol{B}\),

In general, Equations 2 and 16, along with their respective initial conditions given by by Equations 3 and 17, are generally referred to as the *equations of variation* (see, *e.g.*, Arnold, 1992, pp. 223 – 225), and they constitute a well-posed problem. Hence, the Spatial Propagator \(\mathbf{F} ( \lambda , \boldsymbol{r}_{0} )\) solution exists and is unique for all \(\lambda \) and \(\boldsymbol{r}_{0}\) (see, *e.g.*, Bernstein, 2018, pp. 1193 – 1195, for existence and uniqueness proof).

*For any reference point*\(\boldsymbol{r}_{0}\)

*and all*\(\lambda \)

*, the congruence of scale*\(h_{m}\)

*is a particular solution to the equations of variation consisting of both the integral curve*\(\boldsymbol{r} ( \lambda , \boldsymbol{r}_{0} )\)

*and corresponding Spatial Propagator*\(\mathbf{F} ( \lambda , \mathbf{r}_{0} )\)

*. The congruence may be interpreted as a local phase-space of scale*\(h_{m}\)

*for the field-line connectivity consisting of elements*\(\boldsymbol{r} ( \lambda , \boldsymbol{r}_{0} )\)

*and*\(\mathbf{F} ( \lambda , \mathbf{r}_{0} )\)

*.*With respect to the global Cartesian coordinates, these phase space elements are represented by a three-component vector and \(3 \times 3\)-component matrix,

### 2.4 General Representations of the Spatial Propagator

There are two fundamental representation categories of the Spatial Propagator \(\mathbf{F} ( \lambda , \boldsymbol{r}_{0} )\): the matrix representation with respect to a particular basis set \(\hat{\boldsymbol{e}}_{i}\), and the representation with respect to the connectivity parameter \(\lambda \).

First, we consider the basis representation of the congruence; that is the field line \(\boldsymbol{r} ( \lambda , \boldsymbol{r}_{0} )\) and associated Spatial Propagator \(\mathbf{F} ( \lambda , \boldsymbol{r}_{0} )\) are, respectively, a three-component vector and \(3 \times 3\) matrix representation *constructed with respect to a particular basis set*. Fundamentally, the matrix representation of the congruence *follows from* the vector field \(\boldsymbol{B} ( \boldsymbol{r} )\) and its gradient matrix \(\nabla \boldsymbol{B} ( \boldsymbol{r} )\); that is constructing \(\boldsymbol{B}\) and \(\nabla \boldsymbol{B}\) with respect to Cartesian basis vectors lead to the Cartesian representation of the congruence: Equation 19.

^{4}basis vectors,

*we impose the condition that the Spatial Propagator*\(\mathbf{F} ( \lambda , \boldsymbol{r}_{0} )\)

*transform as a tensor under general coordinate transformations*.

*e.g.*, Kobayashi and Nomizu, 1963, pp. 143 – 144). The components of the matrix representation of the covariant differential of the vector field with respect to general local curvilinear coordinates \(q^{i}\) is given by

*e.g.*, Tassev and Sevcheva, 2017, for application in spherical–polar orthonormal coordinates). In Appendices B.2 and B.3 we develop explicit matrix representations of the covariant differential of the vector field \(\nabla \boldsymbol{B} ( \boldsymbol{r} )\) with respect to the common spherical–polar bases used by the solar and space-physics community (see,

*e.g.*, Tassev and Sevcheva, 2017).

Second, we consider the connectivity parameter representation of the congruence. For a given vector field \(\boldsymbol{B} ( \boldsymbol{r} )\), the congruence solution \(\boldsymbol{r} ( \lambda , \boldsymbol{r}_{0} )\) and \(\mathbf{F} ( \lambda , \boldsymbol{r}_{0} )\) to Equations 2 and 16 with respect to \(\lambda \) is said to be given in the *natural representation*. That is the connectivity parameter \(\lambda \) has units of arc length per vector field magnitude, and the congruence elements, \(\boldsymbol{r} ( \lambda , \boldsymbol{r}_{0} )\) and \(\mathbf{F} ( \lambda , \boldsymbol{r}_{0} )\), reflect this functional dependence.

*e.g.*, Tassev and Sevcheva, 2017; Scott, Pontin, and Hornig, 2017, for application).

In the application to solar magnetic fields, constructing the congruence solution from data (*e.g.**Solar Dynamics Observatory*/*Atmospheric Imaging Assembly* 171 Å images) requires the solutions to Equations 8 and 24, with initial conditions 7 and 25, respectively. From this perspective, we have decoupled the magnetic-field strength estimation from the field-line trajectory estimation in the construction of the Spatial Propagator.

## 3 Geometric Deformation of a Congruence

Geometry describes the measurable lengths and angles associated with the system configuration. The value of the congruence formalism is that all of the geometric behavior of a local bundle of field lines is contained within a single integration of Equations 2 and 16, along with their respective initial conditions 3 and 17. The field line \(\boldsymbol{r} ( \lambda , \boldsymbol{r}_{0} )\) and corresponding Spatial Propagator \(\mathbf{F} ( \lambda , \boldsymbol{r}_{0} )\) implicitly incorporate all geometric information of the local bundle, which follows naturally from the vector magnetic-field structure: \(\boldsymbol{B}\) and \(\nabla \boldsymbol{B}\). The generic geometric configuration of the congruence may be described by a combination of dilation (Section 3.1), stretch, rotation (Section 3.2), and gradients in the connectivity structure (Section 3.3).

We recall that by assumption the vector field components are smooth functions. Hence, by standard theorems of existence, uniqueness, and extension for ordinary differential equations (see, *e.g.*, Hirsch and Smale, 1974; Arnold, 1992; Taylor, 1996), each field-line reference point \(\boldsymbol{r}_{0} \in \Omega _{0}\) is mapped smoothly and uniquely to the point \(\boldsymbol{r} = \boldsymbol{r} \left ( \lambda , \boldsymbol{r}_{0} \right ) \in \Omega _{\lambda }\) (see Figure 1); that is the neighborhood volume \(\Omega _{0}\) is mapped, smoothly and uniquely, to the neighborhood volume \(\Omega _{\lambda }\). Hence, *the geometric deformation of the local congruence is reflected in the deformation of the initial volume*\(\Omega _{0}\)*by propagation along the congruence into* \(\Omega _{\lambda }\).

### 3.1 Congruence Dilation and the Determinant of the Spatial Propagator

The dilation of a congruence is the compression/expansion of the constituent field lines, and it is completely described by the determinant of the \(3 \times 3\), non-singular (invertible) matrix representation of the Spatial Propagator \(\mathbf{F} ( \lambda , \boldsymbol{r}_{0} )\). This type of geometric deformation is quantified by the dilation of a volume \(\Omega _{\lambda } \subseteq \mathbb{R}^{3}\) propagated through each \(\lambda \) along the congruence. In this section, we make explicit the functional dependence of the deformed volume on both the connectivity parameter \(\lambda \) and the reference point \(\boldsymbol{r}_{0}\), by denoting \(\Omega _{\lambda } = \Omega ( \lambda , \boldsymbol{r}_{0} )\).

We may choose differential-reference-shift vectors to coincide with the Cartesian coordinate differentials \({\mathrm{d}} \boldsymbol{h}^{i} \equiv {\mathrm{d}} x^{i} \hat{\boldsymbol{e}}_{i}\), and hence write the local signed differential volume element at the reference point \({\mathrm{d}} \Omega ( 0, \boldsymbol{r}_{0} ) = {\mathrm{d}} ^{3} \boldsymbol{x}\).

*e.g.*, Nickerson, Spencer, and Steenrod, 1959, pp. 95 – 97, for a standard treatment). From a geometric perspective, the determinant of the \(3 \times 3\) matrix representation of \(\mathbf{F} ( \lambda , \boldsymbol{r}_{0} )\) acts as the Jacobian in a change of basis for the signed volume element propagated along the congruence (see Figure 3).

The dilation of the total volume \(\Omega ( \lambda , \boldsymbol{r}_{0} )\) by propagation along the congruence is therefore completely determined by the determinant of the Spatial Propagator \(\operatorname{det} \mathbf{F} ( \lambda , \boldsymbol{r} _{0} )\) at each \(\lambda \). Geometrically, the total volume \(\Omega ( \lambda , \boldsymbol{r}_{0} )\) decreases (volumetric compression) for \(0 < \operatorname{det} \mathbf{F} ( \lambda , \boldsymbol{r}_{0} ) < 1\), remains constant (isochoric propagation) for \(\operatorname{det} \mathbf{F} ( \lambda , \boldsymbol{r}_{0} ) = 1\), or increases (volumetric expansion) for \(\operatorname{det} \mathbf{F} ( \lambda , \boldsymbol{r}_{0} ) > 1\), respectively.

We remark, \(\operatorname{det}\mathbf{F} ( \lambda , \boldsymbol{r}_{0} ) < 0\) corresponds to a mapping from a proper right-handed basis set, to a left-handed basis set. Such an \(\mathbf{F} ( \lambda , \boldsymbol{r}_{0} )\) is discontinuous and requires singularities in the vector field \(\boldsymbol{B}\), which we do not consider in this work.

*we identify*\(\operatorname{det} \mathbf{F} ( \lambda , \boldsymbol{r}_{0} ) = 1\)

*for all*\(\lambda \)

*and every*\(\boldsymbol{r}_{0}\)

*as the simplest topological invariant of a congruence generated by a smooth magnetic vector field. This true topological invariant reflects the divergence-free condition.*

### 3.2 Congruence Stretch and Rotation and the Singular Value Decomposition of the Spatial Propagator

We seek a geometric description in which the orthogonality of the reference frame is preserved under rotation and the lengths of the direction vectors are scaled. This description initially suggests the \(\lambda \)-evolution of the eigen-decomposition of the Spatial Propagator matrix \(\mathbf{F}\), in which the eigenvalues and eigenvectors play the role of scale factors and basis directions. The eigenvalues for a general, non-singular, \(3 \times 3\) matrix \(\mathbf{F}\) are determined by a cubic polynomial with real coefficients, the roots of which may be real or complex-valued depending on the sign of the classical cubic discriminant. In the case of non-negative cubic discriminant, the eigenvalues are all real (perhaps with multiplicity \(> 1\)), the geometric interpretation is scaling along linearly independent eigenvectors (since \(\mathbf{F}\) is a non-singular matrix). However, *mutual orthogonality of the eigenvectors follows only in the special case that the Spatial Propagator matrix is symmetric*, \(\mathbf{F} = \mathbf{F}^{T}\) (where \(\mathbf{F}^{T}\) denotes the matrix transpose). Moreover, a negative cubic discriminant corresponds to complex-valued eigenvalues and eigenvectors, the geometric interpretation of which is not a stretch and rotation of the propagated volume. Hence, the eigen-decomposition of the Spatial Propagator \(\mathbf{F}\) does not provide the correct stretch and rotation description.

*change in the length*of propagated vectors \(\boldsymbol{v} \left ( \lambda , \boldsymbol{r}_{0} \right )\) within the propagated volume. The length of a vector is described by the norm

Equation 35 is a real-valued function of \(\lambda \) that involves the symmetric matrix \(\mathbf{F}^{T} \cdot \mathbf{F}\), as opposed to the Spatial Propagator \(\mathbf{F}\) alone. By a standard theorem of linear algebra (see, *e.g.*, Halmos, 1958, Section 79), the eigenvalues of the symmetric matrix \(\mathbf{F}^{T} \cdot \mathbf{F}\) are all real (possibly with multiplicity \(> 1\)), and the eigenvectors are everywhere mutually orthogonal. More generally, the symmetric matrices \(\mathbf{F}^{T} \cdot \mathbf{F}\) and \(\mathbf{F} \cdot \mathbf{F} ^{T}\) have identical eigenvalues, as well as orthogonal, albeit different, eigenvector bases. Hence, for a general, non-singular Spatial Propagator \(\mathbf{F}\), the real eigenvalues and real orthogonal eigenvectors of the symmetric matrices \(\mathbf{F} \cdot \mathbf{F} ^{T}\), respectively \(\mathbf{F}^{T} \cdot \mathbf{F}\), may be used to infer indirectly the geometric stretch and rotation interpretation of \(\mathbf{F}\).

*singular value decomposition*(SVD), given by

*e.g.*Bernstein, 2018, pp. 555 – 558). The diagonal entries of the matrix \(\mathbf{P} ( \lambda , \boldsymbol{r}_{0} )\) are non-negative functions \(\sigma _{\alpha } ( \lambda , \boldsymbol{r}_{0} )\) for \(\alpha = 1, 2, 3\), given by

For each reference point \(\boldsymbol{r}_{0}\) and fixed \(\lambda \), the set of values \(\sigma _{\alpha }\) for \(\alpha = 1, 2, 3\) are called the *singular values* of \(\mathbf{F} ( \lambda , \boldsymbol{r} _{0} )\), and the corresponding set of vectors \(\hat{{\boldsymbol{l}}}_{\alpha }\) and \({\hat{\boldsymbol{r}}}_{\alpha }\) are called the *singular vectors* of \(\mathbf{F} ( \lambda , \boldsymbol{r}_{0} )\). The anisotropic stretch and rigid-body rotation of the propagated volume, equivalently the congruence geometry, is completely determined by the singular values and singular vectors of the Spatial Propagator \(\mathbf{F} ( \lambda , \boldsymbol{r}_{0} )\). Geometrically, an orthogonal frame \({\hat{\boldsymbol{r}}} _{\alpha }\) in the reference volume \(\Omega _{0}\) is rotated into the orthogonal frame \({\hat{\boldsymbol{l}}}_{\alpha }\) in the propagated volume \(\Omega _{\lambda }\) while the vector lengths are simultaneously scaled by the corresponding \(\sigma _{\alpha }\), as shown in Figure 4.

*polar decomposition*of the Spatial Propagator. Since \(\mathbf{R}_{l} ( \lambda , \boldsymbol{r}_{0} )\) is an orthogonal matrix (\(\mathbf{R}^{-1} = \mathbf{R}^{T}\), where superscript “\(T\)” denotes the matrix transpose) for each \(\boldsymbol{r}_{0}\) and all \(\lambda \), then \(\mathbf{R}_{l}^{T} ( \lambda , \boldsymbol{r}_{0} ) \cdot \mathbf{R}_{l} ( \lambda , \boldsymbol{r}_{0} ) = \mathbf{I}\), where \(\mathbf{I}\) is the identity. Hence, we may rewrite Equation 36 as

*left-polar decomposition*and

*right-polar decomposition*, respectively, of the Spatial Propagator (see,

*e.g.*, Halmos, 1958, Section 83). Unlike the general Spatial Propagator \(\mathbf{F} ( \lambda , \boldsymbol{r}_{0} )\) itself, the matrices \(\mathbf{V} ( \lambda , \boldsymbol{r}_{0} )\) and \(\mathbf{U} ( \lambda , \boldsymbol{r}_{0} )\), denoted, respectively, the

*left-stretch*and

*right-stretch*, are symmetric, positive definite, \(3 \times 3\) matrices defined by

*e.g.*, Halmos, 1958, Section 79). Since the matrices \(\mathbf{V} ( \lambda , \boldsymbol{r}_{0} )\) and \(\mathbf{U} ( \lambda , \boldsymbol{r}_{0} )\) are symmetric, there exist orthogonal eigen-bases \({\hat{\boldsymbol{l}}}_{\alpha } ( \lambda , \boldsymbol{r}_{0} )\) and \({\hat{\boldsymbol{r}}} _{\alpha } ( \lambda , \boldsymbol{r}_{0} )\), with respect to which the matrix representations are

*the symmetric matrices*\(\mathbf{V} ( \lambda , \boldsymbol{r}_{0} )\)

*and*\(\mathbf{U} ( \lambda , \boldsymbol{r}_{0} )\)

*, and orthogonal matrix*\(\mathbf{R} ( \lambda , \boldsymbol{r}_{0} )\)

*, respectively, characterize the smooth anisotropic stretch and rigid-body rotation of the propagated volume along the congruence.*

The basis \({\hat{\boldsymbol{l}}}_{\alpha } ( \lambda , \boldsymbol{r}_{0} )\) describes the singular axes of the stretched and rotated state \(\Omega _{\lambda }\). Similarly, for any \(\lambda \) the basis \({\hat{\boldsymbol{r}}}_{\alpha } ( \lambda , \boldsymbol{r} _{0} )\) identify the singular axes of the stretched but un-rotated reference state \(\Omega _{0}\). Hence, for any \(\lambda \) the basis \({\hat{\boldsymbol{r}}}_{\alpha } ( \lambda , \boldsymbol{r} _{0} )\) are the singular directions of the reference state \(\Omega _{0}\). We note, however, a spherical reference state \(\Omega _{0}\) has no preferred directions; this follows from the fact that the initial condition \(\mathbf{F} ( 0, \boldsymbol{r}_{0} ) = \mathbf{I}\) possesses a single degenerate singular value (eigenvalue) \(\sigma = 1\) of multiplicity 3. The resolution of this is that a unique, non-degenerate, orthogonal basis \({\hat{\boldsymbol{r}}}_{\alpha } ( \lambda , \boldsymbol{r}_{0} )\) in the reference state \(\Omega _{0}\) only exists and *follows from* a Spatial Propagator \(\mathbf{F} ( \lambda , \boldsymbol{r}_{0} )\) with non-degenerate singular values.

*directly dependent*on the current distribution. Consider a quasi-static magnetic field \(\boldsymbol{B} ( \boldsymbol{r} )\) with a non-trivial current distribution \(\boldsymbol{J} ( \boldsymbol{r} )\) that satisfies Ampere’s Law: \(\mu _{0} \boldsymbol{J} ( \boldsymbol{r} ) = \nabla \times \boldsymbol{B} ( \boldsymbol{r} )\). The covariant differential of the vector field \(\nabla \boldsymbol{B}\) may be decomposed into symmetric and anti-symmetric parts,

Since the left-stretch \(\mathbf{V} ( \lambda , \boldsymbol{r}_{0} )\) and right-stretch \(\mathbf{U} ( \lambda , \boldsymbol{r}_{0} )\) matrices are symmetric, through Equation 49, the rotation matrix \(\mathbf{R} ( \lambda , \boldsymbol{r}_{0} )\) implicitly requires a non-trivial anti-symmetric part of the covariant differential \(\nabla \boldsymbol{B} ( \boldsymbol{r} )\), equivalently a non-trivial (parallel) current distribution \(\boldsymbol{J} ( \boldsymbol{r} )\). Moreover, under reasonable conditions we identify a class of magnetic fields \(\boldsymbol{B} ( \boldsymbol{r} )\) for which the rotation matrix \(\mathbf{R} ( \lambda , \boldsymbol{r}_{0} )\) depends *only* on the parallel current distribution \(\boldsymbol{J} ( \boldsymbol{r} )\). Geometrically, the congruence rotation describes the twist of all neighboring integral curves about a central axis field line, and is therefore a measure of the twist helicity. The relationship between the Spatial Propagator, rotation matrix, (twist-) magnetic helicity, and (parallel) current distribution is beyond the scope of this work.

### 3.3 Quasi-Separatrix Layers and the \(Q\)-Factor from the Spatial Propagator

We have shown in the previous Sections 3.1 and 3.2 that the congruence geometry is equivalent to the kinematics of a volume undergoing dilation, stretch deformation, and rigid-body rotation under the action of the Spatial Propagator \(\mathbf{F} ( \lambda , \boldsymbol{r}_{0} )\). Hence, a separatrix surface reflects the total-collapse of the 3D volume into a 2D surface as it is propagated by the congruence. The quasi-separatrix layer is similarly identified by an “extreme” kinematic deformation of the 3D volume; “extreme” being a subjective term and requiring a (user-defined) quantified threshold.

We remark, recalling from Section 3.1 that the divergence-free condition for vector magnetic fields leads to the topological invariant \(\operatorname{det} \mathbf{F} ( \lambda , \boldsymbol{r}_{0} ) = 1\). Geometrically, the volumetric kinematics are incompressible;^{5} that is, the 3D shape deforms under the action of \(\mathbf{F} ( \lambda , \boldsymbol{r}_{0} )\) while the total volume remains constant. Hence, the separatrix surfaces in a vector magnetic field are described by infinite stretch deformation of the volume.

- i)
If any one singular value becomes zero, \(\sigma _{i} \to 0\), then the other two correspondingly diverge to infinity, \(\sigma _{j}, \sigma _{k} \to \infty \); geometrically, the 3D field-line bundle has collapsed into a 2D surface.

- ii)
If any two of the singular values become zero, \(\sigma _{i}, \sigma _{j} \to 0\), then the remaining one correspondingly diverges to infinity, \(\sigma _{k} \to \infty \); geometrically, the 3D field-line bundle has collapsed into a 1D line.

- iii)
The divergence-free condition prevents all three singular values from approaching zero simultaneously; geometrically, the 3D incompressible field-line bundle cannot collapse to a 0D point.

For example, in the natural representation the fan surface and spine line(s) associated with a magnetic null are identified in the limit as \(\vert \lambda \vert \to \infty \) by the vanishing of one and two singular values, respectively; the sign of \(\lambda \) depends on the field-line polarity. More generally, however, the identification of other non-trivial topological features that exhibit the vanishing of one, or multiple, singular values at finite and/or infinite \(\vert \lambda \vert \) remains an open question.

Truly infinite singular values \(\sigma _{\alpha } \to \infty \) determine 2D separatrix surfaces and 1D separator lines. “Extreme” singular values identify quasi-separatrix layers (QSLs); that is, *e.g.*, \(\sigma _{i} \leq 1/\epsilon \) and correspondingly \(\sigma _{j}, \sigma _{k} \geq \epsilon \), for some user-defined threshold \(\epsilon \gg 1\). Analogous “extreme” threshold arguments may be made around separator lines.

Up to this point, we have investigated the geometric meaning, consequences, and relations between the singular values of the full-3D Spatial Propagator in various limiting cases. 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. For the remainder of this section, we demonstrate the dimensional reduction of the full-3D Spatial Propagator to describe the 2D squashing of the congruence. In particular, we illustrate a simple construction of the popular squashing factor \(Q\) (Titov, Hornig, and Démoulin, 2002; Titov, 2007), from the Spatial Propagator. We note, our aim here is *neither* a reformulation, nor a more efficient computation of the squashing factor \(Q\) over that available in the open literature (see, *e.g.*, Tassev and Sevcheva, 2017). Rather the purpose is simply to show exactly how the popular \(Q\)-value may be derived from, and therefore fits within, the general Spatial Propagator framework.

Throughout this article we have assumed that the vector magnetic field \(\boldsymbol{B} ( \boldsymbol{r} )\) has smooth component functions, Equation 1, that may be described with respect to a single global Cartesian coordinate chart. Hence the congruence solution components, Equation 19, described with respect to the same global Cartesian coordinate chart, are also smooth, single-valued functions of the connectivity parameter \(\lambda \), and reference condition coordinates \(\boldsymbol{r}_{0}\). Furthermore, by construction, this framework is valid for any global coordinate chart that covers the system (see Appendix B.1 for general coordinate chart formulations, and Appendices B.2 and B.3 for spherical–polar formulations).

Recall from Section 2.1 that we identify \(\lambda = 0\) with the reference point \(\boldsymbol{r}_{0}\), corresponding to the beginning of the congruence, and we identify the *fixed, finite* value \(\lambda = L\) with \(\boldsymbol{r} = \boldsymbol{r} ( L, \boldsymbol{r}_{0} )\), corresponding to the end of the congruence. In general, we take the reference point \(\boldsymbol{r}_{0}\) and end point \(\boldsymbol{r} ( L, \boldsymbol{r}_{0} )\) on the system boundary.^{6} From this perspective, *each particular field-line solution with*\(L > 0\)*provides a unique connectivity map between disjoint points on the system boundary, and the Spatial Propagator quantifies the local geometric organization of this connectivity map within its characteristic scale*.

*not*tangent to their respective boundary surface. However, we may project the \(\boldsymbol{v}_{\alpha } ( L , \boldsymbol{r}_{0} )\) onto the tangent boundary plane (see Figure 6)

*et al.*, 1996) is given by

We remark that *the line-tying condition does not imply a fixed Spatial Propagator; only the proper sub-matrices and their combinatorics forming the squashing factor*\(Q\)*are preserved.*

The \(Q\)-Map (Titov, 2007), identifies the separatrix and QSL field structures. However, on their own, these regions offer only the *possible* current-sheet formation sites in the field structure. Whether or not electromagnetic stresses accumulate resulting in subsequent current-sheet formation depends on the details of energization and stress injection; *e.g.* separatrix and QSL structures undergoing rigid-body motion will not develop a current sheet (Aulanier, Pariat, and Démoulin, 2005; Aulanier *et al.*, 2010; Janvier *et al.*, 2014); whereas more general motions, such as shearing or twisting motions, will develop a current sheet unstable to reconnection (Aulanier *et al.*, 2006; Effenberger *et al.*, 2011; Janvier *et al.*, 2013).

## 4 Conclusion and Future Applications

In this article we introduce a generalized field-line connectivity phase space associated with the vector magnetic field in which the geometric and topological features of the system are made explicit. The fundamental assumption is that the vector magnetic field is *a priori* smooth everywhere. The basic elements are the field line and its linearized variation, the Spatial Propagator: Equation 19. Equations 2 and 16, with initial conditions from Equations 3 and 17, provide a direct formulation of these phase-space elements in terms of the vector magnetic field and its spatial derivatives. Furthermore, the field line and Spatial Propagator are constructed with respect to general curvilinear coordinates and the equivalence class of general affine parameterizations.

The geometric interpretation is that the Spatial Propagator characterizes the organization of the local bundle of field lines. Since the vector field is everywhere smooth, so too are the field-line and Spatial Propagator solutions smooth and unique. The geometric organization of the local bundle is completely equivalent to a kinematic description of a volume centered on the particular field-line solution and undergoing deformation by transport along the field. This deformation kinematics is characterized by volumetric dilation, anisotropic stretch, and rotation. The volumetric dilation (expansion/compression) is completely described by the determinant of the Spatial Propagator: Equation 30. For the vector magnetic field, the determinant of the Spatial Propagator is a topological invariant everywhere equal to unity, Equation 34, which reflects the divergence-free condition. The anisotropic stretch and rotation kinematics are described by the singular values and singular vectors of the Spatial Propagator. The singular values, Equation 37, characterize the general anisotropic stretch of the congruence volume. The congruence rotation is a simple rigid-body rotation kinematics between the orthonormal principal-direction bases: Equation 43, from the reference volume \(\Omega _{0}\) to the propagated volume \(\Omega _{\lambda }\).

Extreme singular values identify QSLs within the system; true separatrix surfaces and separator lines within the system are identified in the limiting cases of one, or two, zero singular values, respectively. Moreover, the \(Q\)-factor is simply constructed from analysis of the particular sub-matrix of the Spatial Propagator obtained by removing the column corresponding to the coordinate direction parallel to the launch-surface normal and row corresponding to the coordinate direction parallel to the target-surface normal.

This magnetic connectivity phase-space framework opens up extensive directions in geometric and topological analysis of vector magnetic fields. For example, in future work we will relax the *a priori* smooth vector magnetic-field assumption in order to analyze both existing singular structures such as current sheets and their formation. Moreover, the magnetic helicity may be decomposed into twist and writhe components (Moffatt and Ricca, 1992); in future efforts we will show that, accounting for relative shearing of the volume, twist helicity may be described with the congruence rotation, while the writhe helicity is related to the embedding of the field-line trajectory and propagation of the reference shift vector in 3D space.

In the present article the field-line bundle and Spatial Propagator are presented and analyzed from a geometric perspective. Since the field lines are curves (*i.e.**spatial positions*) in a three-dimensional space whose tangent vector is everywhere parallel to the magnetic-field vector, evolutionary dynamics (ideal or otherwise) of the field-line bundle are often described by imposing the frozen-in condition (*e.g.* infinite plasma conductivity) somewhere within the system domain (typically at the system boundary). This allows one to describe and follow the dynamics, not of the field line itself, but rather of the parcel of plasma to which the field line is connected. In non-ideal systems; the difference between the ideal motion and the actual motion of the field is attributed to resistive slipping; this is the origin of “slip-running reconnection” and other resistive slip phenomenology. In follow-up analysis, we use this formalism to describe the dynamics of a field-line bundle and associated Spatial Propagator without reference to plasma conductivity, or material parcels, but rather with respect to pure electric and magnetic fields; such a description necessarily requires a treatment of the full four-dimensional electromagnetic-field tensor.

*e.g.*, Jackson, 1999, Section 11.9). The field line generalizes to a 4-vector flow field \(\phi ^{\mu } ( \lambda ; t_{0}, \boldsymbol{r}_{0} )\), where the spatial components are identified with the canonical notion of a field line. In addition, the propagator generalizes to a second-order mixed tensor field \(F^{ \mu }{}_{\nu } ( t, \boldsymbol{x} )\) where the spatial components are identified with the Spatial Propagator described in this article. Moreover, we will show that expanding these four-dimensional generalizations into explicit spatial and temporal components recovers the definition of a magnetic-field line as three

*spatial constraint*equations,

*time-dynamic*equations for the field-line components,

*e.g.*coronal heating, active-region stability, flares, and CME initiation) in future work.

## Footnotes

- 1.
We remark, unless specifically noted otherwise, throughout this article we work with respect to the standard Cartesian coordinates \(\lbrace x, y, z \rbrace \) and orthonormal Cartesian unit basis vectors \(\lbrace \hat{\boldsymbol{e}}_{x}, \hat{\boldsymbol{e}}_{y}, \hat{\boldsymbol{e}}_{z} \rbrace \).

- 2.
The function \(f\) must be at least \(C^{1}\) differentiable. The positivity condition preserves field-line orientation. Furthermore, a solenoidal vector field \(\nabla \cdot \boldsymbol{B} = 0\) with \(\boldsymbol{X} = \boldsymbol{B} / f\) leads to a constraint \(\nabla \cdot \boldsymbol{X} + \boldsymbol{X} \cdot \nabla f = 0\); then choosing a function \(f\) such that the vector field \(\boldsymbol{X}\) is everywhere tangent to the level surfaces of \(f\) (

*i.e.*\(\boldsymbol{X} \cdot \nabla f = 0\)) preserves the solenoidal property to the vector field \(\boldsymbol{X}\). Furthermore, including a temporal dependence on \(f\) imposes a constraint equation on the \(\boldsymbol{X}\)-evolution to preserve consistency with \(\boldsymbol{B}\). - 3.
The value of the constant \(b\) represents a simple relabeling of the reference point on the particular field-line solution;

*i.e.*\(\boldsymbol{r}_{0} \mapsto \boldsymbol{r}_{b}\). - 4.
We reserve the hat-notation for strictly

*orthonormal*basis vectors. - 5.
Compressible vector fields are outside the scope of this work.

- 6.
In solar- and coronal-physics applications, the system boundary is typically taken to be the photosphere-photosphere, or photosphere-source surface ansatz.

- 7.
The subscript labels 1 – 6 are not unique.

- 8.
More generally, the determinant of any \(n \times n\) matrix may be similarly written in terms of the \(S_{n}\) permutation group.

## Notes

### Acknowledgements

This material is based upon work supported by the National Aeronautics and Space Administration under Grant No. NNX15AJ66G and 80NSSC18K1553 issued through the NASA Shared Services Center (NSSC). Any opinions, findings, and conclusions or recommendations expressed in this article are those of the authors and do not necessarily reflect the views of the National Aeronautics and Space Administration. Furthermore, J.K. Edmondson would like to thank L.A. Fisk, D.S. Bernstein, D.T. Welling, B.J. Lynch, B.L. Alterman, N. Gallant, and N. Furbush for invaluable comments and insightful discussions. Finally, both J.K. Edmondson and P. Démoulin would like to thank the organizers of the 2017 SHINE conference in Saint Sauveur, in particular G. de Nolfo and N. Lugaz, where this collaboration began.

### Disclosure of Potential Conflict of Interest

The authors declare that they have no conflicts of interest.

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