Absolute instability of the boundary-layer flows due to rotating a spheroid

This paper explores local absolute instability in the boundary layer flow over two distinct families of rotating spheroids (prolate and oblate). While convective instability was established in earlier work by Samad and Garrett [1], this study delves into the potential occurrence of local absolute instability. Some results of local convective instability under the assumption of stationary vortices are reproduced for a more comprehensive investigation. The analysis considers viscous and streamline curvature effects, demonstrating that the localized mean flow within the boundary layer over either family of the rotating spheroid is absolutely unstable for each fixed value of the eccentricity parameter $e\in [0,0.8]$. For certain combinations of Reynolds number $Re$ and azimuthal wave number $\beta$, a third branch (Branch 3) of the dispersion relation intersects Branch 1 at a pinch point, indicating absolute instability. Neutral curves depict regions that are absolutely unstable, while below critical Reynolds numbers, the region is either convectively unstable or stable. The paper also illustrates the effect of increasing eccentricity on spatial branches within both convectively and absolutely unstable regions. From lower to moderate latitudes, the stabilizing effect of $e$ on the onset of absolute instability is robust for the prolate family and almost negligible for the oblate family. At high latitudes of the prolate spheroid, the stabilizing effect of $e$ is fainter but persists until close to the equator. Conversely, at high latitudes of the oblate spheroid, the stabilizing effect of $e$ is more pronounced. The paper discusses the implications of the parallel flow assumption employed in the analyses.