Inertial waves and modes excited by the libration of a rotating cube

We report experimental measurements of the flow in a cubic container submitted to a longitudinal libration, i.e. a rotation modulated in time. Velocity fields in a vertical and a horizontal plane are measured in the librating frame using a corotating particle image velocimetry system. When the libration frequency $\sigma_0$ is smaller than twice the mean rotation rate $\Omega_0$, inertial waves can propagate in the interior of the fluid. At arbitrary excitation frequencies $\sigma_0<2\Omega_0$, the oscillating flow shows two contributions: (i) a basic flow induced by the libration motion, and (ii) inertial wave beams propagating obliquely upward and downward from the horizontal edges of the cube. In addition to these two contributions, inertial modes may also be excited at some specific resonant frequencies. We characterize in particular the resonance of the mode of lowest order compatible with the symmetries of the forcing, noted [2,1,+]. By comparing the measured flow fields to the expected inviscid inertial modes computed numerically [L.R.M. Maas, Fluid Dyn. Res. \textbf{33}, 373 (2003)], we show that only a subset of inertial modes, matching the symmetries of the forcing, can be excited by the libration.

💡 Research Summary

The paper presents a comprehensive experimental investigation of the flow generated inside a rotating cubic container that is subjected to longitudinal libration – a periodic modulation of its rotation rate. Using a corotating particle‑image‑velocimetry (PIV) system, the authors acquire two‑dimensional velocity fields in both a vertical (x‑z) and a horizontal (x‑y) plane while the cube rotates at a mean angular velocity Ω₀ and oscillates with a small amplitude at frequency σ₀.

The authors first establish the condition for the existence of inertial waves: σ₀ must be smaller than twice the background rotation (σ₀ < 2Ω₀). When this inequality holds, the measured flow can be decomposed into three distinct contributions. (i) A “basic” libration‑induced flow that follows the imposed angular oscillation and is essentially a potential flow modified by Ekman pumping in the thin boundary layers. (ii) Oblique inertial‑wave beams that emanate from the four horizontal edges of the cube and propagate upward and downward at an angle θ given by sin θ = σ₀/(2Ω₀). The beams are clearly visible in the PIV snapshots, their width and amplitude decreasing as σ₀ approaches the cut‑off 2Ω₀, and they experience viscous attenuation consistent with a Reynolds number based on the Ekman layer thickness of order 10⁴. (iii) At discrete resonant frequencies, standing inertial modes are excited.

To identify which modes can be resonantly forced, the authors invoke the symmetry of the cubic geometry and of the libration forcing. The cube possesses three orthogonal mirror planes and a four‑fold rotational symmetry about the vertical axis; the libration preserves the vertical axis and the 90° rotational symmetry. Consequently, only inertial modes whose spatial structure respects these symmetries can receive energy from the forcing. The authors compare their measurements with the inviscid eigenmodes computed by Maas (Fluid Dyn. Res. 33, 373, 2003), which are labelled by a triplet