Classical NLFFF extrapolation methods

The force-free equations define the field we want, but not how to compute it.

Classical NLFFF methods usually represent $\mathbf{B}$ on a numerical grid and search for a field that:

• matches the measured boundary;
• reduces the Lorentz-force residual;
• keeps $\nabla \cdot \mathbf{B}$ small.

They matter here because they are the standard comparison for NF2. NF2 is a different numerical route through the same inverse problem, not a different physics model.

Following Wiegelmann and Sakurai, the common families are below.

1. Grad-Rubin methods

The force-free equations can be written:

$$\nabla \times \mathbf{B}=\alpha \mathbf{B},\mathbf{B}\cdot \nabla \alpha =0.$$

The second equation says $\alpha$ is constant along each field line.

A Grad-Rubin scheme uses that structure directly:

1. Given $\mathbf{B}$, trace field lines and propagate $\alpha$ from the boundary into the volume.
2. Given that current distribution, update $\mathbf{B}$.

This is physically meaningful because $\alpha$ follows magnetic connectivity.

The catch is boundary consistency. In practice, $\alpha$ can only be prescribed cleanly on one magnetic polarity, and real magnetograms are noisy and not exactly force-free.

2. Optimisation methods

Optimisation methods define one functional measuring departure from force-free and divergence-free behaviour:

$$\mathcal{L}={\int }_V[w_f\frac{|(\nabla \times \mathbf{B})\times \mathbf{B}{|}^2}{|\mathbf{B}{|}^2}+w_d|\nabla \cdot \mathbf{B}{|}^2]dV.$$

Read the two terms plainly:

• the first punishes Lorentz force;
• the second punishes magnetic divergence, i.e. fake monopoles.

If $\mathcal{L}=0$, the field is exactly force-free and divergence-free.

The field is evolved in artificial time, downhill in this functional, while the bottom boundary is held fixed to the magnetogram. This family is the direct conceptual bridge to Classical optimisation and the NF2 loss: same residuals, different representation.

3. Magnetofrictional / MHD relaxation methods

Magnetofrictional methods introduce an artificial velocity proportional to the Lorentz force:

$$\mathbf{v}=\frac{1}{\nu }\mathbf{J}\times \mathbf{B}.$$

The field is then evolved with the induction equation:

$$\frac{\partial \mathbf{B}}{\partial t}=\nabla \times (\mathbf{v}\times \mathbf{B}).$$

The friction removes energy from the system, relaxing an initial field, often a potential field, toward a lower-force equilibrium where:

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

This is a numerical relaxation, not a faithful simulation of coronal dynamics. The final state can depend on the initial field, driving, and numerical diffusion.

4. Boundary-integral / Green’s-function methods

Boundary-integral methods construct the field in the volume from boundary data using an integral kernel.

This works naturally for potential fields, where the problem reduces to Laplace’s equation with a known Green’s function.

For nonlinear force-free fields, the currents vary through the volume, so the kernel is no longer simple. The method becomes much less direct and is less central for this project.

The shared difficulty

Every method inherits the same physical mismatch:

boundary measured in the photosphere
model intended for the low-beta corona

The photosphere is not perfectly force-free. Preprocessing can nudge the magnetogram toward force-free consistency, but that changes the input data and must be reported.

NF2 does not escape this problem. It moves it into the training objective through a soft boundary loss and tunable physics weights.

Related

• Classical optimisation and the NF2 loss
• NLFFF quality metrics
• Force-free and NLFFF coronal fields
• Coronal magnetic-field extrapolation as an inverse problem
• NF2 and physics-informed neural coronal extrapolation
• Photosphere versus corona force-free assumption