Literature Review Equations and definitions

Recurring equations and notation for the solar/NF2 literature review.

Magnetic field and current

Magnetostatic Ampere law:

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

Potential field

A potential field is current-free:

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

This is the baseline low-energy extrapolation.

Solenoidal constraint

Magnetic fields should satisfy:

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

This is Gauss’s law for magnetism / no magnetic monopoles.

Force-free / NLFFF condition

Force-free means the Lorentz force vanishes:

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

Substituting for $\mathbf{J}$ gives:

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

Often:

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

where NLFFF allows $\alpha$ to vary spatially.

PINN / NF2 schematic loss

A useful schematic loss is:

$$\mathcal{L}={\lambda }_b{\mathcal{L}}_{\mathrm{boundary}}+{\lambda }_{ff}{\mathcal{L}}_{\mathrm{force}-\mathrm{free}}+{\lambda }_{div}{\mathcal{L}}_{\mathrm{div}}.$$

Magnetic energy

$$E={\int }_V\frac{B^2}{2{\mu }_0}dV,E_{\mathrm{free}}=E_{\mathrm{NLFFF}}-E_{\mathrm{pot}}.$$

Boundary-data notation

For the HMI SHARP CEA boundary used in the local NF2 project, the practical component mapping is:

(Bx, By, Bz) = (Bp, -Bt, Br)

This belongs in the methods/reproduction part of the write-up more than the broad literature review, but it is worth remembering because sign mistakes in vector magnetogram work are extremely easy.

Diagnostic wording

Use “residual” for the leftover equation error after substituting the model field into the equation. Use “loss” for the scalar quantity optimised during training. These are related but not identical.

Related

• Solar Physics Equations and definitions
• Literature Review Key concepts
• draft