Magnetic energy and free energy in active regions

Magnetic energy is the useful currency of coronal modelling. It is what active regions store and what flares and CMEs can release.

It is also a sharp diagnostic for an extrapolation: did the model capture real non-potential structure, or did it mostly reproduce the potential-field baseline?

Total magnetic energy

The magnetic energy stored in a volume $V$ is:

$$E_{\text{mag}}=\frac{1}{2{\mu }_0}{\int }_V|\mathbf{B}{|}^{2}dV.$$

Plainly: each bit of field carries energy density $|\mathbf{B}{|}^2/2{\mu }_0$, and we integrate that over the coronal box.

The potential-field floor

For a fixed normal flux distribution $B_z$ on the boundary, the potential field is the unique lowest-energy field consistent with that boundary.

Adding currents can only increase the energy, provided the same boundary flux is held fixed. So the potential-field energy $E_{\text{pot}}$ sets the floor.

Free magnetic energy

The free magnetic energy is the excess above that floor:

$$E_{\text{free}}=E_{\text{NLFFF}}-E_{\text{pot}}\ge 0.$$

This is the energy that could in principle be released while keeping the boundary flux fixed. It lives in the currents: shear, twist, and non-potential structure.

A current-free field has:

$$E_{\text{free}}=0.$$

Why it is useful for NF2

Free energy cuts both ways.

If NF2 gives almost no excess above the potential baseline, the nonlinear model has not added much physics. It may have learned a smooth current-free field while spending much more computation.

If NF2 gives a large free energy, that is only credible if the other diagnostics agree. A field can inflate its energy by overfitting noisy photospheric structure or carrying spurious currents.

So free energy should be read together with:

• boundary agreement;
• force-freeness;
• divergence control;
• topology and connectivity.

A caution on divergence

Energy estimates are contaminated by divergence error. If $\nabla \cdot \mathbf{B}\ne 0$, part of the computed $|\mathbf{B}{|}^2$ belongs to a non-physical component.

That can bias $E_{\text{mag}}$ and therefore $E_{\text{free}}$. This is why divergence control is not cosmetic. A field with fake monopoles gives untrustworthy energies.

Magnetic helicity is a related non-potential diagnostic and is also relevant for AR 11158.

Related

• Potential field extrapolation
• NLFFF quality metrics
• Force-free and NLFFF coronal fields
• NF2 diagnostics for physical credibility
• NOAA AR 11158 as an NF2 benchmark