  The magneto-rotational decay instability (MRDI) of thin Keplerian discs
threaded by poloidal magnetic fields is introduced and studied. The linear
magnetohydrodynamic problem decouples into eigenvalue problems for in-plane
slow- and fast- Alfv'een-Coriolis (AC), and vertical magnetosonic (MS)
eigenmodes. The magnetorotational instability (MRI) is composed of a discrete
number of unstable slow AC eigenmodes that is determined for each radius by the
local beta. In the vicinity of the first beta threshold a parent MRI eigenmode
together with a stable AC eigenmode (either slow or fast) and a stable MS
eigenmode form a resonant triad. The three-wave MRDI relies on the nonlinear
saturation of the parent MRI mode and the exponential growth of two daughter
linearly stable waves, slow-AC and MS modes with an effective growth rate that
is comparable to that of the parent MRI. If, however, the role of the AC
daughter wave is played by a stable fast mode, all three modes remain bounded.
