Force-free magnetic field

A force-free magnetic field obeys $(\nabla \times \mathbf{B})\times \mathbf{B}=0$, so its electric current is locally parallel to the magnetic field.

A force-free magnetic field is linear when the proportionality between its curl and itself is a spatial constant. The curl of the curl identity and zero divergence give the vector Helmholtz equation ${\nabla }^2\mathbf{B}+{\lambda }^2\mathbf{B}=0$. Here $\lambda$ has inverse-length units; specifying a constant current-to-field coefficient with another normalization changes its numerical definition.

A linear force-free magnetic field independent of $y$ can be written $\mathbf{B}=(-A_z,\lambda A,A_x)$. Choosing $A=(B_0/k)\sin (kx)e^{-qz}$ gives a decaying arcade with $k=n\pi /a$, $n\ge 1$, and $q=\sqrt{k^2-{\lambda }^2}>0$. Both horizontal components vanish at $x=0,a$. A fixed horizontal mode requires $|\lambda |<k$; if the mode number is unrestricted, some sufficiently high mode exists for every finite real $\lambda$. Its magnetic field line projections satisfy $A=\text{constant}$ and $dy/dx=\lambda /q$ within each cell.

The general ideal magnetic-energy balance has outward-surface contribution ${\mu }_0^{-1}\oint [(u\cdot B)B-B^{2}u]\cdot dS$ and volume contribution $-\int u\cdot (J\times B)dV$. The latter vanishes for a force-free magnetic field. For an axisymmetric boundary rotating tangentially, $u\cdot B=\Omega f(\psi )$ and the directed flux of a thin tube is $2\pi d\psi$. Pair its exit and entry endpoints to obtain the displayed energy injection, counting each boundary endpoint once. Equal endpoint angular velocities inject no magnetic energy.

Writing $\nabla \times \mathbf{B}=(4\pi /c)\alpha \mathbf{B}$ defines the force-free parameter $\alpha$. Since the divergence of a curl vanishes and $\nabla \cdot \mathbf{B}=0$, it satisfies $\mathbf{B}\cdot \nabla \alpha =0$.

A cylindrical force-free magnetic field depends only on cylindrical radius and has current everywhere parallel to its magnetic field. Regularity on the axis often forces its radial component to vanish.