Selection of the radial value of the WKBJ instability threshold

Determine which value of the radius-dependent maximum unstable wavenumber k_max should be used in the WKBJ estimates of the unstable-wavenumber range, mode count, and dispersion relation for the wall-bounded rotating flow with power-law angular velocity.

Background

For the wall-bounded incompressible rotating flow, the WKBJ approximation estimates the instability range using a local maximum wavenumber k_max that varies with cylindrical radius. The authors note that this variation makes the approximate predictions non-unique: one may use mean values or derive upper and lower bounds from the extrema of the relevant equilibrium functions. Resolving which choice is appropriate would improve the quantitative interpretation of the WKBJ estimates and their comparison with the exact global dispersion relation.

References

These estimates are necessarily rough, since $k_{\rm max}=\dfrac{\sqrt{\rho_0'g_{\rm eff}/\rho_0-(\Omega2)'}{v_{\rm A}$ depends on radius. In the present case, $k_{\rm max}=\dfrac{\sqrt{-2b}{v_{\rm A}{-b}$, so it is not obvious which value should be used.

— Instabilities in Cylindrical Geometry Using the Minimalist Approach: Formalism and Rotational Instabilities  (2609.02240 - Vlahakis, 2 Sep 2026) in Section 3.1, “Approximate Dispersion Relation Using WKBJ”