Linear force-free magnetic field ($\nabla \times \mathbf{B}=\lambda \mathbf{B},\lambda \text{ constant}$)

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.