Force-free and NLFFF coronal fields

A force-free magnetic field is a field where the Lorentz force vanishes:

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

Using

$$\mathbf{J}=\frac{1}{{\mu }_0}\nabla \times \mathbf{B},$$

this becomes

$$(\nabla \times \mathbf{B})\times \mathbf{B}=0.$$

The field must also be solenoidal:

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

Together, these are the core equations behind NLFFF extrapolation.

Physical meaning

$\mathbf{J}\times \mathbf{B}=0$ means the current density allows follows the magnetic field lines or that current is parallel or anti-parallel to the magnetic field. Current is allowed, but it cannot go in a direction sideways across the field.

So the curl of the field can be written as:

$$\nabla \times \mathbf{B}=\alpha \mathbf{B},$$

where $\alpha$ is the force-free parameter. It measures field-aligned current per unit magnetic field.

The hierarchy

The difference between force-free models is how $\alpha$ behaves:

Model	$\alpha$	Meaning
Potential field	$\alpha =0$	no current
Linear force-free field	$\alpha =\text{const}$	uniform twist
Nonlinear force-free field (NLFFF)	$\alpha =\alpha (\mathbf{x})$	twist and current can vary between field lines

In an NLFFF field, $\alpha$ can vary through the volume, but it is constant along each field line. That follows from $\nabla \cdot \mathbf{B}=0$ and is derived in From MHD to the force-free equations.

Why NLFFF is useful

Active regions contain twist, shear, electric currents, and free magnetic energy. A potential field cannot represent those because it has no current.

NLFFF is the middle ground: much richer than a potential field, but still simpler and cheaper than full time-dependent MHD.

Why it is difficult

NLFFF extrapolation is hard because:

• the equations are nonlinear;
• magnetogram data are noisy;
• the photosphere is not exactly force-free;
• the side and top boundaries are weakly constrained;
• different numerical methods can produce different 3D fields from similar boundary data.

Why this matters for NF2

NF2 and physics-informed neural coronal extrapolation is best understood as a neural optimisation strategy for this same NLFFF target. It should be judged by physical diagnostics, not only by training loss or visually pleasing loops.

Related

• Photosphere versus corona force-free assumption
• Residuals in NF2 and PINNs
• Solenoidal magnetic fields and no magnetic monopoles
• Classical NLFFF extrapolation methods