Rayleigh centrifugal stability criterion (${\kappa }^2=r^{-3}(l^2)\prime$)

For inviscid cylindrical rotation without a magnetic field, an axisymmetric radial interchange conserves each parcel's specific angular momentum $l=r^2\Omega$. Its restoring coefficient is ${\kappa }^2=r^{-3}d(l^2)/dr$. Increasing $l^2$ gives centrifugal stability; decreasing $l^2$ gives an exponentially growing interchange mode when radial displacement is allowed by the boundaries. A vanishing derivative is marginal, not a strictly restoring condition.

// Target: astrophysics.bigb