Asymptotic theory of resonant flow in a spheroidal cavity driven by latitudinal libration

We consider a homogeneous fluid of viscosity $\nu $ confined within an oblate spheroidal cavity, ${x}^{2} / {a}^{2} + {y}^{2} / {a}^{2} + {z}^{2} / ({a}^{2} (1\ensuremath{-} {\mathscr{E}}^{2} ))= 1$, with eccentricity $0\lt \mathscr{E}\lt 1$. The spheroidal container rotates rapidly with an angular velocity ${\mbit{\Omega} }_{0} $, which is fixed in an inertial frame and defines a small Ekman number $E= \nu / ({a}^{2} \vert {\mbit{\Omega} }_{0} \vert )$, and undergoes weak latitudinal libration with frequency $\hat {\omega } \vert {\mbit{\Omega} }_{0} \vert $ and amplitude $\mathit{Po}\vert {\mbit{\Omega} }_{0} \vert $, where $\mathit{Po}$ is the Poincaré number quantifying the strength of Poincaré force resulting from latitudinal libration. We investigate, via both asymptotic and numerical analysis, fluid motion in the spheroidal cavity driven by latitudinal libration. When $\vert \hat {\omega } \ensuremath{-} 2/ (2\ensuremath{-} {\mathscr{E}}^{2} )\vert \gg O({E}^{1/ 2} )$, an asymptotic solution for $E\ll 1$ and $\mathit{Po}\ll 1$ in oblate spheroidal coordinates satisfying the no-slip boundary condition is derived for a spheroidal cavity of arbitrary eccentricity without making any prior assumptions about the spatial–temporal structure of the librating flow. In this case, the librationally driven flow is non-axisymmetric with amplitude $O(\mathit{Po})$, and the role of the viscous boundary layer is primarily passive such that the flow satisfies the no-slip boundary condition. When $\vert \hat {\omega } \ensuremath{-} 2/ (2\ensuremath{-} {\mathscr{E}}^{2} )\vert \ll O({E}^{1/ 2} )$, the librationally driven flow is also non-axisymmetric but latitudinal libration resonates with a spheroidal inertial mode that is in the form of an azimuthally travelling wave in the retrograde direction. The amplitude of the flow becomes $O(\mathit{Po}/ {E}^{1/ 2} )$ at $E\ll 1$ and the role of the viscous boundary layer becomes active in determining the key property of the flow. An asymptotic solution for $E\ll 1$ describing the librationally resonant flow is also derived for an oblate spheroidal cavity of arbitrary eccentricity. Three-dimensional direct numerical simulation in an oblate spheroidal cavity is performed to demonstrate that, in both the non-resonant and resonant cases, a satisfactory agreement is achieved between the asymptotic solution and numerical simulation at $E\ll 1$.