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

Internal shear layers generated by the longitudinal libration of the inner core in a spherical shell rotating at a rate Ω * are analysed asymptotically and numerically. The forcing frequency is chosen as √ 2Ω * such that the layers issued from the inner core boundary at the critical latitude in the form of concentrated conical beams draw a simple rectangular pattern in meridional cross-sections. The asymptotic structure of the internal shear layers is described by extending the self-similar solution known for open domains to closed domains where reflections on the boundaries occur. The periodic nature of the ray path ensures that the internal shear layers remain localised around the periodic orbit. The solution obtained by summing infinitely many cycles is found to converge. The asymptotic predictions are compared to direct numerical results obtained for Ekman number as low as E = 10 −10. The agreement between the asymptotic predictions and numerical results is shown to improve as the Ekman decreases. The scalings E 1/12 for the amplitude and E 1/2 for the dissipation rate predicted by the asymptotic theory are recovered numerically. Since the self-similar solution is singular on the axis, a new local asymptotic solution is derived close to the axis and is also validated numerically. This study demonstrates that, in the limit of vanishing Ekman numbers and for particular frequencies, the main features of the flow generated by a librating inner core are obtained by propagating through the spherical shell the self-similar solution generated by the singularity at the critical latitude on the inner core.