Solar Physics Equations and definitions

Useful equations and definitions for the solar/NF2 project.

Magnetic field and current

Ampere’s law in magnetostatics:

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

Solenoidal constraint

No magnetic monopoles:

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

See Solenoidal magnetic fields and no magnetic monopoles.

Potential field

Current-free field:

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

See Potential field extrapolation.

Force-free / NLFFF condition

Lorentz force vanishes:

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

Using $\mathbf{J}={\mu }_0^{-1}\nabla \times \mathbf{B}$:

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

Often written as:

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

For NLFFF, $\alpha$ can vary in space. See Force-free and NLFFF coronal fields.

Magnetic energy

$$E={\int }_V\frac{B^2}{2{\mu }_0}dV.$$

Free energy relative to potential field:

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

See Magnetic energy and free energy in active regions.

Plasma beta

$$\beta =\frac{p_{\mathrm{gas}}}{p_{\mathrm{mag}}}=\frac{2{\mu }_{0}p}{B^2}.$$

Low $\beta$ supports the force-free approximation. See Photosphere versus corona force-free assumption.

NF2/PINN schematic loss

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

See NF2 and physics-informed neural coronal extrapolation.

Parent / nearby maps

• Solar Physics index
• Solar Physics Key concepts