Magnetic bending in an incompressible disk

In the steady state of the horizontally invariant magnetized shearing-sheet equations, let $B_x(\pm H)=\pm B_{+}$ and $B_y(\pm H)=0$. For $q>0$ and $B_z\ne 0$, the symmetric equilibrium hasThis follows by combining azimuthal induction with radial magnetic tension balance to obtain $B_x^{\prime \prime }+K^2{B}_x=0$. Nonzero $B_{+}$ requires $\sin (KH)\ne 0$; even homogeneous additions at $\cos (KH)=0$ are excluded by imposing midplane symmetry.

For nonzero imposed surface inclination, magnetic bending in an incompressible disk is nonmonotonic when $KH>\pi /2$: the first maximum of $|B_x|$ occurs at $z=\pi /(2K)<H$, followed by a decrease. This is exactly the finite-thickness magnetorotational instability criterion for the same slab. The zero-inclination equilibrium can still be unstable, so nonmonotonic bending is a property of the forced profile, rather than a condition on every possible unstable equilibrium.