Internal shear layers in librating spherical shells: the case of periodic characteristic paths

Internal shear layers generated by the longitudinal libration of inner core in a spherical shell rotating at rate $\varOmega ^*$ are analysed asymptotically and numerically. The forcing frequency is chosen as $\sqrt {2}\varOmega such that issued from critical latitude form concentrated conical beams draw simple rectangular pattern meridional cross-sections. asymptotic structure internal described extending self-similar solution known for open domains to closed where reflections on boundaries occur. periodic ray path ensures remain localised around it. Asymptotic solutions both main beam along line weaker secondary perpendicular it obtained. predictions compared with direct numerical results obtained Ekman numbers low $E=10^{-10}$ . agreement between improves number decreases. scalings $E^{1/12}$ $E^{1/4}$ amplitudes beams, respectively, recovered Since singular axis, new local derived close axis also validated This study demonstrates that, limit vanishing particular frequencies, features flow librating propagating through singularity core.