The non-resonant response of fluid in a rapidly rotating sphere undergoing longitudinal libration

We investigate the problem of oscillatory flow of a homogeneous fluid with viscosity $\nu $ in a fluid-filled sphere of radius $a$ that rotates rapidly about a fixed axis with angular velocity ${\Omega }_{0} $ and that undergoes weak longitudinal libration with amplitude $\epsilon {\Omega }_{0} $ and frequency $\hat {\omega } {\Omega }_{0} $, where $\epsilon $ is the Poincaré number with $\epsilon \ll 1$ and $\hat {\omega } $ is dimensionless frequency with $0\lt \hat {\omega } \lt 2$. Three different methods are employed in this investigation: (i) asymptotic analysis at small Ekman numbers $E$ defined as $E= \nu / ({a}^{2} {\Omega }_{0} )$; (ii) linear numerical analysis using a spectral method; and (iii) nonlinear direct numerical simulation using a finite-element method. A satisfactory agreement among the three different sets of solutions is achieved when $E\leq 1{0}^{- 4} $. It is shown that the flow amplitude $\vert \boldsymbol{u}\vert $ is nearly independent of both the Ekman number $E$ and the libration frequency $\hat {\omega } $, always obeying the asymptotic scaling $\vert \boldsymbol{u}\vert = O(\epsilon )$ even though various spherical inertial modes are excited by longitudinal libration at different libration frequencies $\hat {\omega } $. Consequently, resonances do not occur in this system even when $\hat {\omega } $ is at the characteristic value of an inertial mode. It is also shown that the pressure difference along the axis of rotation is anomalous: this quantity reaches a sharp peak when $\hat {\omega } $ approaches a characteristic value. In contrast, the pressure difference measured at other places in the sphere, such as in the equatorial plane, and the volume-integrated kinetic energy are nearly independent of both the Ekman number $E$ and the libration frequency $\hat {\omega } $. Absence of resonances in a fluid-filled sphere forced by longitudinal libration is explained through the special properties of the analytical solution that satisfies the no-slip boundary condition and is valid for $E\ll 1$ and $\epsilon \ll 1$.