From MHD to the force-free equations

The force-free equations are often written down as if they are the starting point:

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

They are not fundamental. They are what the MHD momentum equation becomes in the static, low-$\beta$ corona.

This derivation matters because it says exactly which terms are being ignored. That is the honest way to state the model’s regime of validity, especially because the photospheric boundary does not satisfy the same assumptions cleanly.

Start from MHD momentum balance

Single-fluid MHD conserves momentum as:

$$\rho \frac{D\mathbf{v}}{Dt}=-\nabla p+\mathbf{J}\times \mathbf{B}+\rho \mathbf{g}+\mu {\nabla }^2\mathbf{v}.$$

The terms are:

• inertia, $\rho D\mathbf{v}/Dt$;
• gas-pressure force, $-\nabla p$;
• Lorentz force, $\mathbf{J}\times \mathbf{B}$;
• gravity, $\rho \mathbf{g}$;
• viscosity, $\mu {\nabla }^2\mathbf{v}$.

The convective derivative is:

$$\frac{D}{Dt}={\partial }_t+\mathbf{v}\cdot \nabla .$$

We now remove terms using coronal physics.

Quasi-static corona

An active region evolves over hours to days, but magnetic stress can communicate through the corona on an Alfven crossing time:

$${\tau }_A=\frac{L}{v_A}\sim \frac{10^7\text{m}}{10^6{\text{m s}}^{-1}}\sim 10\text{s}.$$

So the field relaxes much faster than the photosphere drives it:

$${\tau }_A\ll {\tau }_{\text{ev}}.$$

That makes the coronal field approximately static at each instant of the slow evolution. We drop inertia and viscosity:

$$0=-\nabla p+\mathbf{J}\times \mathbf{B}+\rho \mathbf{g}.$$

This is now a force-balance problem.

Low beta

The plasma beta is:

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

In the corona, $\beta$ is often much less than one. Magnetic pressure dominates gas pressure, so pressure and gravity are small corrections compared with the Lorentz force.

In the limit $\beta \to 0$, the balance reduces to:

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

This is the force-free condition.

The important caveat is that this is a leading-order approximation. It is best in the low-$\beta$ corona. It is much weaker in the photosphere, where gas pressure, gravity, and flows are not negligible. That mismatch is the root of the boundary problem discussed in Photosphere versus corona force-free assumption.

From force-free to curl form

Use Ampere’s law in magnetostatics:

$$\mathbf{J}=\frac{1}{{\mu }_0}\nabla \times \mathbf{B}.$$

Then the force-free condition becomes:

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

This says the curl of the field is parallel to the field itself. Therefore there is a scalar function $\alpha (\mathbf{x})$ such that:

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

$\alpha$ is the force-free parameter. It measures field-aligned current per unit magnetic field.

Alpha is constant along field lines

This is not an extra assumption. It follows from the solenoidal condition.

Take the divergence of:

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

The divergence of a curl is zero, so:

$$0=\nabla \cdot (\alpha \mathbf{B})=\alpha \nabla \cdot \mathbf{B}+\mathbf{B}\cdot \nabla \alpha .$$

Using $\nabla \cdot \mathbf{B}=0$ gives:

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

So $\alpha$ does not change along a magnetic field line. Different field lines can have different $\alpha$, but a single line carries one value along its length.

This is why Grad-Rubin methods propagate $\alpha$ along field lines from the boundary.

Model hierarchy

Model	$\alpha$	Meaning
Potential	$\alpha =0$	no currents
Linear force-free	$\alpha =\text{const}$	same twist everywhere
Nonlinear force-free	$\alpha$ varies between field lines	local twist, shear, and free energy

Why this matters for NF2

NF2’s physics loss is built from the two surviving equations:

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

This derivation says three useful things.

First, NF2 is solving the static low-$\beta$ limit of MHD, not full MHD. It does not model eruptions, flows, pressure forces, or photospheric dynamics directly.

Second, the hardest mismatch sits at the boundary. The photosphere supplies the magnetogram, but the force-free approximation belongs more naturally to the corona.

Third, the loss is not missing a hidden force-free equation. If the force-free and divergence residuals vanish everywhere, the field is a valid NLFFF solution. If low loss does not give a good physical field, the problem is more likely boundary inconsistency, sampling, optimisation, or diagnostics.

Related

• Force-free and NLFFF coronal fields
• Photosphere versus corona force-free assumption
• Classical NLFFF extrapolation methods
• Classical optimisation and the NF2 loss
• Solenoidal magnetic fields and no magnetic monopoles
• Preprocessing vector magnetograms for NLFFF