Curvature-induced instability of Ekman boundary layers

The linear stability of curved Ekman boundary layers remains poorly understood despite its relevance to rotating geophysical and engineering flows. We investigate the onset of centrifugal instability in an Ekman boundary layer developing over a weakly curved surface, characterized by the dimensionless curvature parameter εc=δE/Rc, where δE is the Ekman boundary—layer thickness and Rc the local radius of curvature. A local weak-curvature formulation incorporating a leading-order model of the centrifugal effect of streamline curvature into the classical Ekman stability equations is developed. The resulting generalized Orr–Sommerfeld/Squire eigenvalue problem is solved using Chebyshev spectral collocation and complemented by an asymptotic analysis of the weak-curvature limit. The numerical results reveal a curvature-induced instability that may be interpreted as a rotation-modified centrifugal instability. The numerical formulation is independently validated in the flat-wall limit by recovering the classical Ekman instability threshold and critical-roll orientation. Optimization over the spanwise wavenumber yields the physically relevant neutral stability boundary. The critical Ekman Reynolds number follows the Görtler–Ekman scaling ReE,c∼εc−1/2, with a fitted exponent of −0.502. The instability also selects a preferred spanwise wavelength proportional to the Ekman boundary–layer thickness. No systematic departure from the weak-curvature scaling is resolved over the parameter range investigated; possible finite-curvature corrections associated with rotational confinement are therefore discussed only as a phenomenological extension. These results demonstrate how streamline curvature and rotation jointly govern centrifugal instability in Ekman boundary layers. Curvature provides the destabilizing mechanism, while rotation controls the confinement and transverse organization of the unstable modes, extending classical centrifugal-instability theory to rotating boundary layers.