Rayleigh discriminant ($\Phi =r^{-3}(r^4{\Omega }^2{)}^{\prime }$)

For inviscid axisymmetric disturbances to a rotating flow, the Rayleigh discriminant measures the radial gradient of squared specific angular momentum.

For axisymmetric inviscid modes between impermeable cylinders in a thin annulus, let $A=2\Omega +r{\Omega }^{\prime }$ and ${\kappa }_{\mathrm{ep}}^2=2\Omega A=r^{-3}(r^4{\Omega }^2{)}^{\prime }$. The linear azimuthal equation gives $u_{\varphi }=-iA{u}_r/\omega$ and the axial equation gives $u_z=k\pi /\omega$, where $\pi$ is the kinematic pressure perturbation. Thin-gap incompressibility implies $\pi =i\omega u_r^{\prime }/k^2$. Eliminating it from the radial equation gives the title equation with $u(R_{<})=u(R_{>})=0$. If a plus sign is used for the potential, its coefficient must be defined as $\Phi =-{\kappa }_{\mathrm{ep}}^2$, the negative of the usual Rayleigh discriminant.

Multiply the narrow-gap centrifugal stability equation by the complex conjugate mode and use integration by parts with zero endpoint radial velocities. This gives the displayed real Rayleigh quotient. Its denominator is positive. Positive Rayleigh discriminant throughout the gap gives positive ${\omega }^2$ for every nonzero mode, excluding exponential axisymmetric growth; nonnegativity permits marginal cases. In the alternative convention $\Phi =-{\kappa }_{\mathrm{ep}}^2$, the numerator is $-k^2\int \Phi |u{|}^{2}dr$, so the stable sign is $\Phi <0$. For rigid rotation, ${\kappa }_{\mathrm{ep}}^2=4{\Omega }^2$ gives ${\omega }_n^2=4{\Omega }^2{k}^2/[k^2+(n\pi /d{)}^2]$.

An inviscid rotating flow is centrifugally stable to axisymmetric disturbances when $(r^2\Omega {)}^2$ is nondecreasing with radius. An outward decrease permits centrifugal instability.