Magnetic topology and field-line tracing

Scalar metrics tell us whether a field obeys the equations. Topology tells us what the field is doing: where it connects, where it twists, where energy is stored, and where reconnection is likely.

For an eruptive region like AR 11158, topology is often the most interesting output.

It is also the easiest place to fool ourselves. A field-line plot can look plausible while the underlying field has poor divergence, weak boundary agreement, or smoothed-away structure. So topology should be extracted carefully and read together with the quantitative metrics.

The field-line equation

A field line is a curve tangent to the magnetic field everywhere. Parametrised by arc length $s$:

$$\frac{d\mathbf{r}}{ds}=\frac{\mathbf{B}(\mathbf{r})}{|\mathbf{B}(\mathbf{r})|}.$$

Tracing a field line means integrating this ODE from a seed point.

In practice, use RK4 or adaptive RK45, with a step size tied to the grid scale or field variation scale. Stop when the line leaves the volume, hits the lower boundary, or reaches a maximum length.

Two details matter:

• Errors accumulate. A small integration error can become a large footpoint error after a long path.
• Seeding is a choice. A regular grid of boundary seeds gives broad connectivity; seeds around flux concentrations or suspected nulls resolve specific structures.

So a topology plot should state how the field lines were seeded and traced.

Why NF2 helps here

NF2 represents the field as a continuous function:

$$\mathbf{B}={\mathcal{N}}_{\theta }(\mathbf{x}).$$

That means $\mathbf{B}$ and its derivatives are available at arbitrary coordinates. A classical grid field needs interpolation between grid cells; NF2 can evaluate the neural field directly.

This is a real advantage of the representation. It does not change the physics, but it can make field-line tracing and derivative-heavy diagnostics cleaner.

Connectivity and footpoint mapping

Trace field lines from one boundary footpoint to the other. This gives a mapping:

$$(x,y)\mapsto (X,Y).$$

The map partitions the lower boundary into flux domains. Smooth regions of the map have stable connectivity. Sharp changes mark topological structure.

Connectivity is one of the most useful checks because it is close to what EUV loops visually suggest, but it is still grounded in the magnetic field model.

Null points

A magnetic null is a point where:

$$\mathbf{B}=0.$$

Near a null, linearise the field:

$$\mathbf{B}\approx M\delta \mathbf{r},$$

where:

$$M_{ij}={\partial }_j{B}_i.$$

Because a solenoidal field has $\nabla \cdot \mathbf{B}=0$,

$$\mathrm{tr}M=0,$$

so the eigenvalues of $M$ sum to zero:

$${\lambda }_1+{\lambda }_2+{\lambda }_3=0.$$

A generic null has one eigenvalue of one sign and two of the other. That gives a 1D spine and a 2D fan surface. Nulls are important because they are natural sites for magnetic reconnection.

Separatrices and separators

Separatrices are surfaces where magnetic connectivity changes discontinuously. Fan surfaces from null points are common examples.

Separators are curves where separatrix surfaces intersect. They are likely places for current sheets and reconnection.

In a numerical field, these structures are sensitive to resolution and divergence error, so they should not be reported without checking field quality.

Quasi-separatrix layers

Real coronal fields often do not have perfect separatrices. Instead, connectivity can change very rapidly but continuously.

These regions are called quasi-separatrix layers (QSLs). They are measured using the squashing factor $Q$ of the footpoint map.

For the footpoint map:

$$D=\left(\begin{array}{cc}\partial X/\partial x& \partial X/\partial y\\ \partial Y/\partial x& \partial Y/\partial y\end{array}\right),$$

the squashing factor is:

$$Q=\frac{\Vert D{\Vert }_F^2}{|\det D|}=\frac{({\partial }_{x}X{)}^2+({\partial }_{y}X{)}^2+({\partial }_{x}Y{)}^2+({\partial }_{y}Y{)}^2}{|{\partial }_{x}X{\partial }_{y}Y-{\partial }_{y}X{\partial }_{x}Y|}.$$

$Q\ge 2$ always. Very large $Q$ marks places where neighbouring field lines separate strongly. These are likely sites for current concentration and reconnection.

QSLs are demanding diagnostics because they depend on both integration and differentiation of the field-line map. Small field errors can become large $Q$ errors.

Flux ropes and shear

A flux rope is a coherent bundle of twisted magnetic field lines.

The twist of a field line can be estimated with:

$$\mathcal{T}=\frac{1}{4\pi }\int \frac{{\mu }_0{J}_{\parallel }}{B}ds=\frac{1}{4\pi }\int \alpha ds,$$

where $\alpha$ is the force-free parameter.

Large twist is relevant for eruptivity. Sheared arcades are related structures where field lines run nearly along the polarity inversion line instead of crossing it cleanly.

Why topology is a strict test

Topology is path-integrated and derivative-sensitive. That makes it useful, but unforgiving.

A small coherent divergence error can change connectivity or create/remove null points. Over-smoothing can reduce ${\sigma }_J$ while dissolving real QSLs, flux ropes, or shear.

So topology should not be used as decoration. A credible topological feature should:

• survive reasonable changes in resolution and seeding;
• appear alongside acceptable divergence and force-free metrics;
• make sense relative to the boundary magnetogram;
• be compared with a potential-field reference;
• coincide with plausible free-energy structure.

That is the useful version of “the loops look right”: a visual feature becomes a measured feature.

Related

• NLFFF quality metrics
• NF2 diagnostics for physical credibility
• Magnetic energy and free energy in active regions
• Solenoidal magnetic fields and no magnetic monopoles
• From MHD to the force-free equations
• NOAA AR 11158 as an NF2 benchmark