TITLE: How can force-free magnetic fields have current parallel to magnetic field?
QUESTION [1 upvotes]: We know that for a force-free magnetic field, $\vec{J} \times \vec{B} = 0$, which means that the current should be parallel to the magnetic field that is creating it. However, from Ampère's law $\vec{\nabla} \times \vec{B} = \mu_{0}\vec{J}$, so I would expect that the current would rotate in the $\hat{\phi}$ direction around the magnetic field.
How is it that the current is parallel to the magnetic field?

REPLY [2 votes]: Which means that the current should be parallel to the magnetic field that is creating it.

I am not sure if this was intentional or just a misphrased statement, but currents are the sources of fields, not the converse.  In the case of a magnetic field-aligned current, there is usually an external source (e.g., planetary magnetic field) and the field-aligned current is locally generated (e.g., magnetic field-aligned beam of one particle species, not both).

However from Maxwell's law $\nabla \times \mathbf{B} = \mu_{o} \mathbf{j}$ so I would expect that the current would rotate in the $\hat{\phi}$ direction around the magnetic field.

Again, recall that the source of the field is something like a current or some net magnetization due to atomic/particle spins.  For this reason, a magnetic field-aligned current will locally cause a perturbation in a background magnetic field, caused by something like a planetary body, that does curl about the background magnetic field.  It would look kind of like the circuit drawing for an inductor, where the "coiled up" part would exist where the magnetic field-aligned current exists.

We know that for a force-free magnetic field $\mathbf{j} \times \mathbf{B} = 0$.  Which means that the current should be parallel to the magnetic field that is creating it.

Returning to the second statement here, you can can construct a scenario where the current is not parallel to the background magnetic field.  That is, let the following be true:  $B_{z} = B_{y} = B_{o}$, $B_{x} = 0$, $j_{z} = j_{y} = j_{o}$, $j_{x} = 0$.  We also linearize so that quantities obey the following $Q = Q_{o} + \delta Q$, where $Q_{o}$ are static terms and $\delta Q$ is a fluctuating term that can depend upon time or space.  In this way, we don't violate Maxwell's equations in the linear limit and still satisfy $\delta \mathbf{j} \times \mathbf{B}_{o} = 0$, i.e., a force-free field without magnetic field-aligned currents.
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