Past exam of the mathematics course of the University of Cambridge 2019 iii Paper 321 3 d Solution Created 2026-10-03 Updated 2026-10-05
Using , the finite-thickness magnetorotational instability criterion is preciselyFor the midplane-symmetric equilibrium with , the absolute radial magnetic field isIts first maximum occurs at . The instability condition puts strictly inside the disc: . The magnitude rises from zero to this maximum, then decreases on a nonempty interval before the surface. Thus the unstable equilibrium exhibits nonmonotonic magnetic bending, as illustrated below.
Monotonic and nonmonotonic magnetic bending
. The symmetric equilibrium profiles have . The weak-field example has an interior maximum and admits a growing magnetic mode.The link between bending and instability concerns a nonzero imposed surface inclination. If , the unbent equilibrium can still have magnetorotational instability; the literal nonmonotonic-bending conclusion does not apply to that degenerate case. Resonances with also require the separate equilibrium analysis in part (b).
