Decaying linear force-free magnetic arcade ($q^2=k^2-{\lambda }^2$)

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.