Elliptical instability in rotating ellipsoidal fluid shells: Applications to the Earth's fluid core

If a circular flow of a contained rotating fluid is perturbed into an elliptical one in such a way that the elliptical streamlines preserve their figures in a laboratory reference frame, the flow becomes elliptically unstable. This type of instability is known to be excited as a result of resonance between the inertial modes of the container. The present work is concerned with the elliptical instability of a thick, ellipsoidal, rotating fluid shell. The aim of this project is twofold. First a theoretical model is derived which describes the dynamics of the elliptical instability in an ellipsoidal fluid shell. The numerical solutions for the pressure field of the inertial modes of the flow are then used to solve the equations of the above model by means of a Galerkin method to obtain the frequency and growth rate of the instability. To gauge the reliability of this approach, the numerical results for a special case are compared with their analytical counterparts and the agreement is to within 0.1% for shells of small ellipticity. Results are presented for several cases, including some where the boundary perturbation is allowed to rotate slowly with respect to the laboratory reference frame. Secondly, experiments are conducted and the elliptical instability as well as some of the inertial modes of a rotating fluid contained in a thick elliptical shell are excited. The velocities of the fluid particles are captured by means of Digital Particle Imaging Velocimetry (DPIV). Plots of the velocity vectors of the inertial modes obtained experimentally are used to identify the theoretical modes. Such investigations are of geophysical interest, since tidal forcing might be sufficient to excite an elliptical instability in the Earth's fluid core as well as in other planetary interiors.