= Global variational form of the Michael criterion
For rigid walls $a<r<b$, a nonzero vertical <wavenumber> $k$ and radial <Lagrangian fluid displacement> $\xi$, set $y=r\xi$ and $\mathcal D=\kappa^2-r(d/dr)(v_A^2/r^2)$. Axisymmetric modes satisfy
$$
\omega^2\left[-\left(\frac{y'}r\right)'+\frac{k^2y}r\right]=\frac{k^2\mathcal D}r y,\qquad y(a)=y(b)=0.
$$
The operator on the left is a positive <self-adjoint operator>. Its <Rayleigh quotient> is $\omega^2=k^2\int\mathcal D|y|^2/r\,dr\big/\int(|y'|^2+k^2|y|^2)/r\,dr$. All mode squares are real; a growing mode exists exactly when the continuous coefficient $\mathcal D$ is negative somewhere. Sufficiency follows by a test function supported in that negative interval and the <compact operator> and <self-adjoint operator> eigenproblem obtained by conjugating the multiplication operator with the inverse square root of the positive differential operator.
// Target: astrophysical-fluid-dynamics.bigb
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