Resonant and non-resonant flows in longitudinally and latitudinally librating spheres

Abstract

We investigate the linear response to longitudinal and latitudinal libration of a rapidly rotating fluid-filled sphere. Asymptotic methods are used to explore the structure of resonant modes in both cases, provided that the nondimensional libration frequency is in the range ω∈[0,2]. High-resolution numerics are then used to map out this entire frequency range, picking out both the resonant peaks as well as the non-resonant troughs in between. The kinetic energy is independent of the Ekman number E at the peaks and scales as E1/2 at the troughs. As the Ekman number is reduced, down to E=10−10 for longitudinal libration and E=10−9 for latitudinal libration, the frequency response also exhibits an increasingly fractal structure, with more and more peaks and troughs emerging. The spacing between peaks is seen to follow an E1/2 self-similarity factor. However, detailed examinations of some of the more prominent troughs shows that their widths follow an E∼0.23 self-similarity factor.