Potential field extrapolation

A potential magnetic field is the simplest coronal extrapolation model. It assumes there is no electric current in the volume:

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

Using magnetostatic Ampere’s law,

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

this means:

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

So a potential field is a current-free field.

Why it is useful

The potential field is the boring baseline: the lowest-complexity magnetic field compatible with the observed normal flux at the boundary.

It is useful because it is mathematically clean, cheap to compute, and gives a reference field to compare against. It is limited because no current means no twist, no shear, and no free magnetic energy.

That is the trade: easy and stable, but physically too simple for eruptive active regions.

Scalar potential form

Since the curl is zero, the field can be written as the gradient of a scalar potential:

$$\mathbf{B}=\nabla \Phi .$$

The magnetic field must also satisfy Gauss’s law for magnetism:

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

Substituting $\mathbf{B}=\nabla \Phi$ gives Laplace’s equation:

$${\nabla }^2\Phi =0.$$

So potential-field extrapolation is a boundary-value problem: solve Laplace’s equation above the photosphere using the observed normal magnetic field as the lower boundary.

Relation to magnetic energy

For a fixed normal flux distribution on the boundary, the potential field is the minimum-energy field. Any field with currents has extra magnetic energy above this floor.

That excess is the free magnetic energy:

$$E_{\mathrm{free}}=E_{\mathrm{NLFFF}}-E_{\mathrm{pot}}.$$

If an NF2 or NLFFF extrapolation barely rises above the potential-field energy, then the nonlinear model has not added much physical structure. It may have learned the floor rather than the interesting currents.

Why this matters for NF2

A potential extrapolation is the first serious baseline for NF2. The question is not just whether NF2 makes nicer-looking field lines. The question is whether it improves on the potential field in ways that survive physical diagnostics:

• lower force-free and divergence errors;
• credible free energy;
• plausible connectivity and topology;
• sensible agreement with the magnetogram.

Related

• Magnetic energy and free energy in active regions
• Force-free and NLFFF coronal fields
• Coronal magnetic-field extrapolation as an inverse problem
• NF2 diagnostics for physical credibility